Step |
Hyp |
Ref |
Expression |
1 |
|
2onn |
|- 2o e. _om |
2 |
|
nnfi |
|- ( 2o e. _om -> 2o e. Fin ) |
3 |
1 2
|
ax-mp |
|- 2o e. Fin |
4 |
|
enfi |
|- ( P ~~ 2o -> ( P e. Fin <-> 2o e. Fin ) ) |
5 |
3 4
|
mpbiri |
|- ( P ~~ 2o -> P e. Fin ) |
6 |
5
|
adantl |
|- ( ( X e. P /\ P ~~ 2o ) -> P e. Fin ) |
7 |
|
diffi |
|- ( P e. Fin -> ( P \ { X } ) e. Fin ) |
8 |
6 7
|
syl |
|- ( ( X e. P /\ P ~~ 2o ) -> ( P \ { X } ) e. Fin ) |
9 |
8
|
cardidd |
|- ( ( X e. P /\ P ~~ 2o ) -> ( card ` ( P \ { X } ) ) ~~ ( P \ { X } ) ) |
10 |
9
|
ensymd |
|- ( ( X e. P /\ P ~~ 2o ) -> ( P \ { X } ) ~~ ( card ` ( P \ { X } ) ) ) |
11 |
|
simpl |
|- ( ( X e. P /\ P ~~ 2o ) -> X e. P ) |
12 |
|
dif1card |
|- ( ( P e. Fin /\ X e. P ) -> ( card ` P ) = suc ( card ` ( P \ { X } ) ) ) |
13 |
6 11 12
|
syl2anc |
|- ( ( X e. P /\ P ~~ 2o ) -> ( card ` P ) = suc ( card ` ( P \ { X } ) ) ) |
14 |
|
cardennn |
|- ( ( P ~~ 2o /\ 2o e. _om ) -> ( card ` P ) = 2o ) |
15 |
1 14
|
mpan2 |
|- ( P ~~ 2o -> ( card ` P ) = 2o ) |
16 |
|
df-2o |
|- 2o = suc 1o |
17 |
15 16
|
eqtrdi |
|- ( P ~~ 2o -> ( card ` P ) = suc 1o ) |
18 |
17
|
adantl |
|- ( ( X e. P /\ P ~~ 2o ) -> ( card ` P ) = suc 1o ) |
19 |
13 18
|
eqtr3d |
|- ( ( X e. P /\ P ~~ 2o ) -> suc ( card ` ( P \ { X } ) ) = suc 1o ) |
20 |
|
suc11reg |
|- ( suc ( card ` ( P \ { X } ) ) = suc 1o <-> ( card ` ( P \ { X } ) ) = 1o ) |
21 |
19 20
|
sylib |
|- ( ( X e. P /\ P ~~ 2o ) -> ( card ` ( P \ { X } ) ) = 1o ) |
22 |
10 21
|
breqtrd |
|- ( ( X e. P /\ P ~~ 2o ) -> ( P \ { X } ) ~~ 1o ) |
23 |
|
en1 |
|- ( ( P \ { X } ) ~~ 1o <-> E. x ( P \ { X } ) = { x } ) |
24 |
22 23
|
sylib |
|- ( ( X e. P /\ P ~~ 2o ) -> E. x ( P \ { X } ) = { x } ) |
25 |
|
simplll |
|- ( ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) /\ X = x ) -> X e. P ) |
26 |
25
|
elexd |
|- ( ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) /\ X = x ) -> X e. _V ) |
27 |
|
simplr |
|- ( ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) /\ X = x ) -> ( P \ { X } ) = { x } ) |
28 |
|
sneqbg |
|- ( X e. P -> ( { X } = { x } <-> X = x ) ) |
29 |
28
|
biimpar |
|- ( ( X e. P /\ X = x ) -> { X } = { x } ) |
30 |
29
|
ad4ant14 |
|- ( ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) /\ X = x ) -> { X } = { x } ) |
31 |
27 30
|
eqtr4d |
|- ( ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) /\ X = x ) -> ( P \ { X } ) = { X } ) |
32 |
31
|
ineq2d |
|- ( ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) /\ X = x ) -> ( { X } i^i ( P \ { X } ) ) = ( { X } i^i { X } ) ) |
33 |
|
disjdif |
|- ( { X } i^i ( P \ { X } ) ) = (/) |
34 |
|
inidm |
|- ( { X } i^i { X } ) = { X } |
35 |
32 33 34
|
3eqtr3g |
|- ( ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) /\ X = x ) -> (/) = { X } ) |
36 |
35
|
eqcomd |
|- ( ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) /\ X = x ) -> { X } = (/) ) |
37 |
|
snprc |
|- ( -. X e. _V <-> { X } = (/) ) |
38 |
36 37
|
sylibr |
|- ( ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) /\ X = x ) -> -. X e. _V ) |
39 |
26 38
|
pm2.65da |
|- ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) -> -. X = x ) |
40 |
39
|
neqned |
|- ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) -> X =/= x ) |
41 |
|
simpr |
|- ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) -> ( P \ { X } ) = { x } ) |
42 |
41
|
unieqd |
|- ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) -> U. ( P \ { X } ) = U. { x } ) |
43 |
|
vex |
|- x e. _V |
44 |
43
|
unisn |
|- U. { x } = x |
45 |
42 44
|
eqtrdi |
|- ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) -> U. ( P \ { X } ) = x ) |
46 |
40 45
|
neeqtrrd |
|- ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) -> X =/= U. ( P \ { X } ) ) |
47 |
46
|
necomd |
|- ( ( ( X e. P /\ P ~~ 2o ) /\ ( P \ { X } ) = { x } ) -> U. ( P \ { X } ) =/= X ) |
48 |
24 47
|
exlimddv |
|- ( ( X e. P /\ P ~~ 2o ) -> U. ( P \ { X } ) =/= X ) |