let p be MP-variable; for A, B being MP-wff holds
( @ p <> 'not' A & @ p <> (#) A & @ p <> A '&' B )
let A, B be MP-wff; ( @ p <> 'not' A & @ p <> (#) A & @ p <> A '&' B )
set e2 = elementary_tree 2;
set e1 = elementary_tree 1;
set e0 = elementary_tree 0;
A1:
dom (@ p) = elementary_tree 0
by FUNCOP_1:13;
A2:
<*0*> in elementary_tree 1
by TARSKI:def 2, TREES_1:51;
dom ((elementary_tree 1) --> [1,0]) = elementary_tree 1
by FUNCOP_1:13;
then
dom ('not' A) = (elementary_tree 1) with-replacement (<*0*>,(dom A))
by A2, TREES_2:def 11;
then
<*0*> in dom ('not' A)
by A2, TREES_1:def 9;
hence
@ p <> 'not' A
by A1, TARSKI:def 1, TREES_1:29; ( @ p <> (#) A & @ p <> A '&' B )
dom ((elementary_tree 1) --> [1,1]) = elementary_tree 1
by FUNCOP_1:13;
then
dom ((#) A) = (elementary_tree 1) with-replacement (<*0*>,(dom A))
by A2, TREES_2:def 11;
then
<*0*> in dom ((#) A)
by A2, TREES_1:def 9;
hence
@ p <> (#) A
by A1, TARSKI:def 1, TREES_1:29; @ p <> A '&' B
set y = ((elementary_tree 2) --> [2,0]) with-replacement (<*0*>,A);
A3:
( <*1*> in elementary_tree 2 & not <*0*> is_a_proper_prefix_of <*1*> )
by TREES_1:28, TREES_1:52;
A4:
( <*0*> in elementary_tree 2 & dom ((elementary_tree 2) --> [2,0]) = elementary_tree 2 )
by FUNCOP_1:13, TREES_1:28;
then
dom (((elementary_tree 2) --> [2,0]) with-replacement (<*0*>,A)) = (dom ((elementary_tree 2) --> [2,0])) with-replacement (<*0*>,(dom A))
by TREES_2:def 11;
then A5:
<*1*> in dom (((elementary_tree 2) --> [2,0]) with-replacement (<*0*>,A))
by A4, A3, TREES_1:def 9;
then
dom (A '&' B) = (dom (((elementary_tree 2) --> [2,0]) with-replacement (<*0*>,A))) with-replacement (<*1*>,(dom B))
by TREES_2:def 11;
then
<*1*> in dom (A '&' B)
by A5, TREES_1:def 9;
hence
@ p <> A '&' B
by A1, TARSKI:def 1, TREES_1:29; verum