Step |
Hyp |
Ref |
Expression |
1 |
|
hmphdis.1 |
|- X = U. J |
2 |
|
dfsn2 |
|- { (/) } = { (/) , (/) } |
3 |
|
indislem |
|- { (/) , ( _I ` A ) } = { (/) , A } |
4 |
|
preq2 |
|- ( ( _I ` A ) = (/) -> { (/) , ( _I ` A ) } = { (/) , (/) } ) |
5 |
4 2
|
eqtr4di |
|- ( ( _I ` A ) = (/) -> { (/) , ( _I ` A ) } = { (/) } ) |
6 |
3 5
|
syl5eqr |
|- ( ( _I ` A ) = (/) -> { (/) , A } = { (/) } ) |
7 |
6
|
breq2d |
|- ( ( _I ` A ) = (/) -> ( J ~= { (/) , A } <-> J ~= { (/) } ) ) |
8 |
7
|
biimpac |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> J ~= { (/) } ) |
9 |
|
hmph0 |
|- ( J ~= { (/) } <-> J = { (/) } ) |
10 |
8 9
|
sylib |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> J = { (/) } ) |
11 |
10
|
unieqd |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> U. J = U. { (/) } ) |
12 |
|
0ex |
|- (/) e. _V |
13 |
12
|
unisn |
|- U. { (/) } = (/) |
14 |
13
|
eqcomi |
|- (/) = U. { (/) } |
15 |
11 1 14
|
3eqtr4g |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> X = (/) ) |
16 |
15
|
preq2d |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> { (/) , X } = { (/) , (/) } ) |
17 |
2 10 16
|
3eqtr4a |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> J = { (/) , X } ) |
18 |
|
hmphen |
|- ( J ~= { (/) , A } -> J ~~ { (/) , A } ) |
19 |
|
necom |
|- ( ( _I ` A ) =/= (/) <-> (/) =/= ( _I ` A ) ) |
20 |
|
fvex |
|- ( _I ` A ) e. _V |
21 |
|
pr2nelem |
|- ( ( (/) e. _V /\ ( _I ` A ) e. _V /\ (/) =/= ( _I ` A ) ) -> { (/) , ( _I ` A ) } ~~ 2o ) |
22 |
12 20 21
|
mp3an12 |
|- ( (/) =/= ( _I ` A ) -> { (/) , ( _I ` A ) } ~~ 2o ) |
23 |
19 22
|
sylbi |
|- ( ( _I ` A ) =/= (/) -> { (/) , ( _I ` A ) } ~~ 2o ) |
24 |
23
|
adantl |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> { (/) , ( _I ` A ) } ~~ 2o ) |
25 |
3 24
|
eqbrtrrid |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> { (/) , A } ~~ 2o ) |
26 |
|
entr |
|- ( ( J ~~ { (/) , A } /\ { (/) , A } ~~ 2o ) -> J ~~ 2o ) |
27 |
18 25 26
|
syl2an2r |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> J ~~ 2o ) |
28 |
|
hmphtop1 |
|- ( J ~= { (/) , A } -> J e. Top ) |
29 |
28
|
adantr |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> J e. Top ) |
30 |
1
|
toptopon |
|- ( J e. Top <-> J e. ( TopOn ` X ) ) |
31 |
29 30
|
sylib |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> J e. ( TopOn ` X ) ) |
32 |
|
en2top |
|- ( J e. ( TopOn ` X ) -> ( J ~~ 2o <-> ( J = { (/) , X } /\ X =/= (/) ) ) ) |
33 |
31 32
|
syl |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> ( J ~~ 2o <-> ( J = { (/) , X } /\ X =/= (/) ) ) ) |
34 |
27 33
|
mpbid |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> ( J = { (/) , X } /\ X =/= (/) ) ) |
35 |
34
|
simpld |
|- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> J = { (/) , X } ) |
36 |
17 35
|
pm2.61dane |
|- ( J ~= { (/) , A } -> J = { (/) , X } ) |