H1a: {{3},{4}} c= sring4_8

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {{3},{4}} or x in sring4_8 )
assume x in {{3},{4}} ; :: thesis: x in sring4_8
then ( x = {3} or x = {4} ) by TARSKI:def 2;
hence x in sring4_8 by ENUMSET1:def 6; :: thesis: verum

end;

H2: {{3},{4}} is Subset-Family of {3,4}

proof

{{3},{4}} c= bool {3,4}

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {{3},{4}} or x in bool {3,4} )
assume x in {{3},{4}} ; :: thesis: x in bool {3,4}
then AAA: ( x = {3} or x = {4} ) by TARSKI:def 2;
( {3} c= {3,4} & {4} c= {3,4} ) by ZFMISC_1:7;
hence x in bool {3,4} by AAA; :: thesis: verum

end;

hence {{3},{4}} is Subset-Family of {3,4} ; :: thesis: verum

end;

H3: union {{3},{4}} = {3,4} by ZFMISC_1:26;
for x being Subset of {3,4} st x in {{3},{4}} holds
( x <> {} & ( for y being Subset of {3,4} holds
( not y in {{3},{4}} or x = y or x misses y ) ) )

proof

let x be Subset of {3,4}; :: thesis: ( x in {{3},{4}} implies ( x <> {} & ( for y being Subset of {3,4} holds
( not y in {{3},{4}} or x = y or x misses y ) ) ) )
assume AA0: x in {{3},{4}} ; :: thesis: ( x <> {} & ( for y being Subset of {3,4} holds
( not y in {{3},{4}} or x = y or x misses y ) ) )
hence x <> {} ; :: thesis: for y being Subset of {3,4} holds
( not y in {{3},{4}} or x = y or x misses y )
for y being Subset of {3,4} holds
( not y in {{3},{4}} or x = y or x misses y )

proof

let y be Subset of {3,4}; :: thesis: ( not y in {{3},{4}} or x = y or x misses y )
assume y in {{3},{4}} ; :: thesis: ( x = y or x misses y )
then ( ( y = {3} or y = {4} ) & ( x = {3} or x = {4} ) ) by AA0, TARSKI:def 2;
hence ( x = y or x misses y ) by LL24; :: thesis: verum

end;

hence for y being Subset of {3,4} holds
( not y in {{3},{4}} or x = y or x misses y ) ; :: thesis: verum

end;

hence ( {{3},{4}} is Subset of sring4_8 & {{3},{4}} is a_partition of {3,4} ) by H1a, H2, H3, EQREL_1:def 4; :: thesis: verum