let a1, a2 be set ; :: thesis: ( ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),{x,y}] in A & [(x 'in' y),{x,y}] in A ) ) & [H,a1] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is being_equality implies a = {(Var1 H'),(Var2 H')} ) & ( H' is being_membership implies a = {(Var1 H'),(Var2 H')} ) & ( H' is negative implies ex b being set st
( a = b & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = union {b,c} & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = (union {b}) \ {(bound_in H')} & [(the_scope_of H'),b] in A ) ) ) ) ) & ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),{x,y}] in A & [(x 'in' y),{x,y}] in A ) ) & [H,a2] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is being_equality implies a = {(Var1 H'),(Var2 H')} ) & ( H' is being_membership implies a = {(Var1 H'),(Var2 H')} ) & ( H' is negative implies ex b being set st
( a = b & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = union {b,c} & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = (union {b}) \ {(bound_in H')} & [(the_scope_of H'),b] in A ) ) ) ) ) implies a1 = a2 )
assume that
A1: ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),H₁(x,y)] in A & [(x 'in' y),H₂(x,y)] in A ) ) & [H,a1] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is being_equality implies a = H₁( Var1 H', Var2 H') ) & ( H' is being_membership implies a = H₂( Var1 H', Var2 H') ) & ( H' is negative implies ex b being set st
( a = H₃(b) & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = H₄(b,c) & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = H₅( bound_in H',b) & [(the_scope_of H'),b] in A ) ) ) ) ) and
A2: ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),H₁(x,y)] in A & [(x 'in' y),H₂(x,y)] in A ) ) & [H,a2] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is being_equality implies a = H₁( Var1 H', Var2 H') ) & ( H' is being_membership implies a = H₂( Var1 H', Var2 H') ) & ( H' is negative implies ex b being set st
( a = H₃(b) & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = H₄(b,c) & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = H₅( bound_in H',b) & [(the_scope_of H'),b] in A ) ) ) ) ) ; :: thesis: a1 = a2
thus a1 = a2 from ZF_MODEL:sch 2(A1, A2); :: thesis: verum