Step |
Hyp |
Ref |
Expression |
1 |
|
diffi |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∖ { 𝑋 } ) ∈ Fin ) |
2 |
|
isfi |
⊢ ( ( 𝐴 ∖ { 𝑋 } ) ∈ Fin ↔ ∃ 𝑚 ∈ ω ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) |
3 |
|
simp3 |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) |
4 |
|
en2sn |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ) → { 𝑋 } ≈ { 𝑚 } ) |
5 |
4
|
3adant3 |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → { 𝑋 } ≈ { 𝑚 } ) |
6 |
|
incom |
⊢ ( ( 𝐴 ∖ { 𝑋 } ) ∩ { 𝑋 } ) = ( { 𝑋 } ∩ ( 𝐴 ∖ { 𝑋 } ) ) |
7 |
|
disjdif |
⊢ ( { 𝑋 } ∩ ( 𝐴 ∖ { 𝑋 } ) ) = ∅ |
8 |
6 7
|
eqtri |
⊢ ( ( 𝐴 ∖ { 𝑋 } ) ∩ { 𝑋 } ) = ∅ |
9 |
8
|
a1i |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( ( 𝐴 ∖ { 𝑋 } ) ∩ { 𝑋 } ) = ∅ ) |
10 |
|
nnord |
⊢ ( 𝑚 ∈ ω → Ord 𝑚 ) |
11 |
|
ordirr |
⊢ ( Ord 𝑚 → ¬ 𝑚 ∈ 𝑚 ) |
12 |
10 11
|
syl |
⊢ ( 𝑚 ∈ ω → ¬ 𝑚 ∈ 𝑚 ) |
13 |
|
disjsn |
⊢ ( ( 𝑚 ∩ { 𝑚 } ) = ∅ ↔ ¬ 𝑚 ∈ 𝑚 ) |
14 |
12 13
|
sylibr |
⊢ ( 𝑚 ∈ ω → ( 𝑚 ∩ { 𝑚 } ) = ∅ ) |
15 |
14
|
3ad2ant2 |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( 𝑚 ∩ { 𝑚 } ) = ∅ ) |
16 |
|
unen |
⊢ ( ( ( ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ∧ { 𝑋 } ≈ { 𝑚 } ) ∧ ( ( ( 𝐴 ∖ { 𝑋 } ) ∩ { 𝑋 } ) = ∅ ∧ ( 𝑚 ∩ { 𝑚 } ) = ∅ ) ) → ( ( 𝐴 ∖ { 𝑋 } ) ∪ { 𝑋 } ) ≈ ( 𝑚 ∪ { 𝑚 } ) ) |
17 |
3 5 9 15 16
|
syl22anc |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( ( 𝐴 ∖ { 𝑋 } ) ∪ { 𝑋 } ) ≈ ( 𝑚 ∪ { 𝑚 } ) ) |
18 |
|
difsnid |
⊢ ( 𝑋 ∈ 𝐴 → ( ( 𝐴 ∖ { 𝑋 } ) ∪ { 𝑋 } ) = 𝐴 ) |
19 |
|
df-suc |
⊢ suc 𝑚 = ( 𝑚 ∪ { 𝑚 } ) |
20 |
19
|
eqcomi |
⊢ ( 𝑚 ∪ { 𝑚 } ) = suc 𝑚 |
21 |
20
|
a1i |
⊢ ( 𝑋 ∈ 𝐴 → ( 𝑚 ∪ { 𝑚 } ) = suc 𝑚 ) |
22 |
18 21
|
breq12d |
⊢ ( 𝑋 ∈ 𝐴 → ( ( ( 𝐴 ∖ { 𝑋 } ) ∪ { 𝑋 } ) ≈ ( 𝑚 ∪ { 𝑚 } ) ↔ 𝐴 ≈ suc 𝑚 ) ) |
23 |
22
|
3ad2ant1 |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( ( ( 𝐴 ∖ { 𝑋 } ) ∪ { 𝑋 } ) ≈ ( 𝑚 ∪ { 𝑚 } ) ↔ 𝐴 ≈ suc 𝑚 ) ) |
24 |
17 23
|
mpbid |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → 𝐴 ≈ suc 𝑚 ) |
25 |
|
peano2 |
⊢ ( 𝑚 ∈ ω → suc 𝑚 ∈ ω ) |
26 |
25
|
3ad2ant2 |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → suc 𝑚 ∈ ω ) |
27 |
|
cardennn |
⊢ ( ( 𝐴 ≈ suc 𝑚 ∧ suc 𝑚 ∈ ω ) → ( card ‘ 𝐴 ) = suc 𝑚 ) |
28 |
24 26 27
|
syl2anc |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( card ‘ 𝐴 ) = suc 𝑚 ) |
29 |
|
cardennn |
⊢ ( ( ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ∧ 𝑚 ∈ ω ) → ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) = 𝑚 ) |
30 |
29
|
ancoms |
⊢ ( ( 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) = 𝑚 ) |
31 |
30
|
3adant1 |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) = 𝑚 ) |
32 |
|
suceq |
⊢ ( ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) = 𝑚 → suc ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) = suc 𝑚 ) |
33 |
31 32
|
syl |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → suc ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) = suc 𝑚 ) |
34 |
28 33
|
eqtr4d |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( card ‘ 𝐴 ) = suc ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) ) |
35 |
34
|
3expib |
⊢ ( 𝑋 ∈ 𝐴 → ( ( 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( card ‘ 𝐴 ) = suc ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) ) ) |
36 |
35
|
com12 |
⊢ ( ( 𝑚 ∈ ω ∧ ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 ) → ( 𝑋 ∈ 𝐴 → ( card ‘ 𝐴 ) = suc ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) ) ) |
37 |
36
|
rexlimiva |
⊢ ( ∃ 𝑚 ∈ ω ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑚 → ( 𝑋 ∈ 𝐴 → ( card ‘ 𝐴 ) = suc ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) ) ) |
38 |
2 37
|
sylbi |
⊢ ( ( 𝐴 ∖ { 𝑋 } ) ∈ Fin → ( 𝑋 ∈ 𝐴 → ( card ‘ 𝐴 ) = suc ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) ) ) |
39 |
1 38
|
syl |
⊢ ( 𝐴 ∈ Fin → ( 𝑋 ∈ 𝐴 → ( card ‘ 𝐴 ) = suc ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) ) ) |
40 |
39
|
imp |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ) → ( card ‘ 𝐴 ) = suc ( card ‘ ( 𝐴 ∖ { 𝑋 } ) ) ) |