Step |
Hyp |
Ref |
Expression |
1 |
|
cardne |
⊢ ( ( card ‘ 𝐵 ) ∈ ( card ‘ 𝐴 ) → ¬ ( card ‘ 𝐵 ) ≈ 𝐴 ) |
2 |
|
ennum |
⊢ ( 𝐴 ≈ 𝐵 → ( 𝐴 ∈ dom card ↔ 𝐵 ∈ dom card ) ) |
3 |
2
|
biimpa |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card ) → 𝐵 ∈ dom card ) |
4 |
|
cardid2 |
⊢ ( 𝐵 ∈ dom card → ( card ‘ 𝐵 ) ≈ 𝐵 ) |
5 |
3 4
|
syl |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card ) → ( card ‘ 𝐵 ) ≈ 𝐵 ) |
6 |
|
ensym |
⊢ ( 𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴 ) |
7 |
6
|
adantr |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card ) → 𝐵 ≈ 𝐴 ) |
8 |
|
entr |
⊢ ( ( ( card ‘ 𝐵 ) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴 ) → ( card ‘ 𝐵 ) ≈ 𝐴 ) |
9 |
5 7 8
|
syl2anc |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card ) → ( card ‘ 𝐵 ) ≈ 𝐴 ) |
10 |
1 9
|
nsyl3 |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card ) → ¬ ( card ‘ 𝐵 ) ∈ ( card ‘ 𝐴 ) ) |
11 |
|
cardon |
⊢ ( card ‘ 𝐴 ) ∈ On |
12 |
|
cardon |
⊢ ( card ‘ 𝐵 ) ∈ On |
13 |
|
ontri1 |
⊢ ( ( ( card ‘ 𝐴 ) ∈ On ∧ ( card ‘ 𝐵 ) ∈ On ) → ( ( card ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ↔ ¬ ( card ‘ 𝐵 ) ∈ ( card ‘ 𝐴 ) ) ) |
14 |
11 12 13
|
mp2an |
⊢ ( ( card ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ↔ ¬ ( card ‘ 𝐵 ) ∈ ( card ‘ 𝐴 ) ) |
15 |
10 14
|
sylibr |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card ) → ( card ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ) |
16 |
|
cardne |
⊢ ( ( card ‘ 𝐴 ) ∈ ( card ‘ 𝐵 ) → ¬ ( card ‘ 𝐴 ) ≈ 𝐵 ) |
17 |
|
cardid2 |
⊢ ( 𝐴 ∈ dom card → ( card ‘ 𝐴 ) ≈ 𝐴 ) |
18 |
|
id |
⊢ ( 𝐴 ≈ 𝐵 → 𝐴 ≈ 𝐵 ) |
19 |
|
entr |
⊢ ( ( ( card ‘ 𝐴 ) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵 ) → ( card ‘ 𝐴 ) ≈ 𝐵 ) |
20 |
17 18 19
|
syl2anr |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card ) → ( card ‘ 𝐴 ) ≈ 𝐵 ) |
21 |
16 20
|
nsyl3 |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card ) → ¬ ( card ‘ 𝐴 ) ∈ ( card ‘ 𝐵 ) ) |
22 |
|
ontri1 |
⊢ ( ( ( card ‘ 𝐵 ) ∈ On ∧ ( card ‘ 𝐴 ) ∈ On ) → ( ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐴 ) ↔ ¬ ( card ‘ 𝐴 ) ∈ ( card ‘ 𝐵 ) ) ) |
23 |
12 11 22
|
mp2an |
⊢ ( ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐴 ) ↔ ¬ ( card ‘ 𝐴 ) ∈ ( card ‘ 𝐵 ) ) |
24 |
21 23
|
sylibr |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card ) → ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐴 ) ) |
25 |
15 24
|
eqssd |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card ) → ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ) |
26 |
|
ndmfv |
⊢ ( ¬ 𝐴 ∈ dom card → ( card ‘ 𝐴 ) = ∅ ) |
27 |
26
|
adantl |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card ) → ( card ‘ 𝐴 ) = ∅ ) |
28 |
2
|
notbid |
⊢ ( 𝐴 ≈ 𝐵 → ( ¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card ) ) |
29 |
28
|
biimpa |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card ) → ¬ 𝐵 ∈ dom card ) |
30 |
|
ndmfv |
⊢ ( ¬ 𝐵 ∈ dom card → ( card ‘ 𝐵 ) = ∅ ) |
31 |
29 30
|
syl |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card ) → ( card ‘ 𝐵 ) = ∅ ) |
32 |
27 31
|
eqtr4d |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card ) → ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ) |
33 |
25 32
|
pm2.61dan |
⊢ ( 𝐴 ≈ 𝐵 → ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ) |