Step |
Hyp |
Ref |
Expression |
1 |
|
df-ne |
⊢ ( ( card ‘ 𝐴 ) ≠ ∅ ↔ ¬ ( card ‘ 𝐴 ) = ∅ ) |
2 |
|
ndmfv |
⊢ ( ¬ 𝐵 ∈ dom card → ( card ‘ 𝐵 ) = ∅ ) |
3 |
|
eqeq1 |
⊢ ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) → ( ( card ‘ 𝐴 ) = ∅ ↔ ( card ‘ 𝐵 ) = ∅ ) ) |
4 |
2 3
|
syl5ibr |
⊢ ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) → ( ¬ 𝐵 ∈ dom card → ( card ‘ 𝐴 ) = ∅ ) ) |
5 |
4
|
con1d |
⊢ ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) → ( ¬ ( card ‘ 𝐴 ) = ∅ → 𝐵 ∈ dom card ) ) |
6 |
5
|
imp |
⊢ ( ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ∧ ¬ ( card ‘ 𝐴 ) = ∅ ) → 𝐵 ∈ dom card ) |
7 |
|
cardid2 |
⊢ ( 𝐵 ∈ dom card → ( card ‘ 𝐵 ) ≈ 𝐵 ) |
8 |
6 7
|
syl |
⊢ ( ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ∧ ¬ ( card ‘ 𝐴 ) = ∅ ) → ( card ‘ 𝐵 ) ≈ 𝐵 ) |
9 |
|
breq2 |
⊢ ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) → ( 𝐴 ≈ ( card ‘ 𝐴 ) ↔ 𝐴 ≈ ( card ‘ 𝐵 ) ) ) |
10 |
|
entr |
⊢ ( ( 𝐴 ≈ ( card ‘ 𝐵 ) ∧ ( card ‘ 𝐵 ) ≈ 𝐵 ) → 𝐴 ≈ 𝐵 ) |
11 |
10
|
ex |
⊢ ( 𝐴 ≈ ( card ‘ 𝐵 ) → ( ( card ‘ 𝐵 ) ≈ 𝐵 → 𝐴 ≈ 𝐵 ) ) |
12 |
9 11
|
syl6bi |
⊢ ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) → ( 𝐴 ≈ ( card ‘ 𝐴 ) → ( ( card ‘ 𝐵 ) ≈ 𝐵 → 𝐴 ≈ 𝐵 ) ) ) |
13 |
|
cardid2 |
⊢ ( 𝐴 ∈ dom card → ( card ‘ 𝐴 ) ≈ 𝐴 ) |
14 |
|
ndmfv |
⊢ ( ¬ 𝐴 ∈ dom card → ( card ‘ 𝐴 ) = ∅ ) |
15 |
13 14
|
nsyl4 |
⊢ ( ¬ ( card ‘ 𝐴 ) = ∅ → ( card ‘ 𝐴 ) ≈ 𝐴 ) |
16 |
15
|
ensymd |
⊢ ( ¬ ( card ‘ 𝐴 ) = ∅ → 𝐴 ≈ ( card ‘ 𝐴 ) ) |
17 |
12 16
|
impel |
⊢ ( ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ∧ ¬ ( card ‘ 𝐴 ) = ∅ ) → ( ( card ‘ 𝐵 ) ≈ 𝐵 → 𝐴 ≈ 𝐵 ) ) |
18 |
8 17
|
mpd |
⊢ ( ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ∧ ¬ ( card ‘ 𝐴 ) = ∅ ) → 𝐴 ≈ 𝐵 ) |
19 |
1 18
|
sylan2b |
⊢ ( ( ( card ‘ 𝐴 ) = ( card ‘ 𝐵 ) ∧ ( card ‘ 𝐴 ) ≠ ∅ ) → 𝐴 ≈ 𝐵 ) |