Step |
Hyp |
Ref |
Expression |
1 |
|
diffi |
|- ( A e. Fin -> ( A \ { X } ) e. Fin ) |
2 |
|
isfi |
|- ( ( A \ { X } ) e. Fin <-> E. m e. _om ( A \ { X } ) ~~ m ) |
3 |
|
simp3 |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> ( A \ { X } ) ~~ m ) |
4 |
|
en2sn |
|- ( ( X e. A /\ m e. _om ) -> { X } ~~ { m } ) |
5 |
4
|
3adant3 |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> { X } ~~ { m } ) |
6 |
|
disjdifr |
|- ( ( A \ { X } ) i^i { X } ) = (/) |
7 |
6
|
a1i |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> ( ( A \ { X } ) i^i { X } ) = (/) ) |
8 |
|
nnord |
|- ( m e. _om -> Ord m ) |
9 |
|
ordirr |
|- ( Ord m -> -. m e. m ) |
10 |
8 9
|
syl |
|- ( m e. _om -> -. m e. m ) |
11 |
|
disjsn |
|- ( ( m i^i { m } ) = (/) <-> -. m e. m ) |
12 |
10 11
|
sylibr |
|- ( m e. _om -> ( m i^i { m } ) = (/) ) |
13 |
12
|
3ad2ant2 |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> ( m i^i { m } ) = (/) ) |
14 |
|
unen |
|- ( ( ( ( A \ { X } ) ~~ m /\ { X } ~~ { m } ) /\ ( ( ( A \ { X } ) i^i { X } ) = (/) /\ ( m i^i { m } ) = (/) ) ) -> ( ( A \ { X } ) u. { X } ) ~~ ( m u. { m } ) ) |
15 |
3 5 7 13 14
|
syl22anc |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> ( ( A \ { X } ) u. { X } ) ~~ ( m u. { m } ) ) |
16 |
|
difsnid |
|- ( X e. A -> ( ( A \ { X } ) u. { X } ) = A ) |
17 |
|
df-suc |
|- suc m = ( m u. { m } ) |
18 |
17
|
eqcomi |
|- ( m u. { m } ) = suc m |
19 |
18
|
a1i |
|- ( X e. A -> ( m u. { m } ) = suc m ) |
20 |
16 19
|
breq12d |
|- ( X e. A -> ( ( ( A \ { X } ) u. { X } ) ~~ ( m u. { m } ) <-> A ~~ suc m ) ) |
21 |
20
|
3ad2ant1 |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> ( ( ( A \ { X } ) u. { X } ) ~~ ( m u. { m } ) <-> A ~~ suc m ) ) |
22 |
15 21
|
mpbid |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> A ~~ suc m ) |
23 |
|
peano2 |
|- ( m e. _om -> suc m e. _om ) |
24 |
23
|
3ad2ant2 |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> suc m e. _om ) |
25 |
|
cardennn |
|- ( ( A ~~ suc m /\ suc m e. _om ) -> ( card ` A ) = suc m ) |
26 |
22 24 25
|
syl2anc |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> ( card ` A ) = suc m ) |
27 |
|
cardennn |
|- ( ( ( A \ { X } ) ~~ m /\ m e. _om ) -> ( card ` ( A \ { X } ) ) = m ) |
28 |
27
|
ancoms |
|- ( ( m e. _om /\ ( A \ { X } ) ~~ m ) -> ( card ` ( A \ { X } ) ) = m ) |
29 |
28
|
3adant1 |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> ( card ` ( A \ { X } ) ) = m ) |
30 |
|
suceq |
|- ( ( card ` ( A \ { X } ) ) = m -> suc ( card ` ( A \ { X } ) ) = suc m ) |
31 |
29 30
|
syl |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> suc ( card ` ( A \ { X } ) ) = suc m ) |
32 |
26 31
|
eqtr4d |
|- ( ( X e. A /\ m e. _om /\ ( A \ { X } ) ~~ m ) -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) |
33 |
32
|
3expib |
|- ( X e. A -> ( ( m e. _om /\ ( A \ { X } ) ~~ m ) -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) ) |
34 |
33
|
com12 |
|- ( ( m e. _om /\ ( A \ { X } ) ~~ m ) -> ( X e. A -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) ) |
35 |
34
|
rexlimiva |
|- ( E. m e. _om ( A \ { X } ) ~~ m -> ( X e. A -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) ) |
36 |
2 35
|
sylbi |
|- ( ( A \ { X } ) e. Fin -> ( X e. A -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) ) |
37 |
1 36
|
syl |
|- ( A e. Fin -> ( X e. A -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) ) |
38 |
37
|
imp |
|- ( ( A e. Fin /\ X e. A ) -> ( card ` A ) = suc ( card ` ( A \ { X } ) ) ) |