Step |
Hyp |
Ref |
Expression |
1 |
|
brwdom |
⊢ ( 𝐵 ∈ dom card → ( 𝐴 ≼* 𝐵 ↔ ( 𝐴 = ∅ ∨ ∃ 𝑥 𝑥 : 𝐵 –onto→ 𝐴 ) ) ) |
2 |
|
0domg |
⊢ ( 𝐵 ∈ dom card → ∅ ≼ 𝐵 ) |
3 |
|
breq1 |
⊢ ( 𝐴 = ∅ → ( 𝐴 ≼ 𝐵 ↔ ∅ ≼ 𝐵 ) ) |
4 |
2 3
|
syl5ibrcom |
⊢ ( 𝐵 ∈ dom card → ( 𝐴 = ∅ → 𝐴 ≼ 𝐵 ) ) |
5 |
|
fodomnum |
⊢ ( 𝐵 ∈ dom card → ( 𝑥 : 𝐵 –onto→ 𝐴 → 𝐴 ≼ 𝐵 ) ) |
6 |
5
|
exlimdv |
⊢ ( 𝐵 ∈ dom card → ( ∃ 𝑥 𝑥 : 𝐵 –onto→ 𝐴 → 𝐴 ≼ 𝐵 ) ) |
7 |
4 6
|
jaod |
⊢ ( 𝐵 ∈ dom card → ( ( 𝐴 = ∅ ∨ ∃ 𝑥 𝑥 : 𝐵 –onto→ 𝐴 ) → 𝐴 ≼ 𝐵 ) ) |
8 |
1 7
|
sylbid |
⊢ ( 𝐵 ∈ dom card → ( 𝐴 ≼* 𝐵 → 𝐴 ≼ 𝐵 ) ) |
9 |
|
domwdom |
⊢ ( 𝐴 ≼ 𝐵 → 𝐴 ≼* 𝐵 ) |
10 |
8 9
|
impbid1 |
⊢ ( 𝐵 ∈ dom card → ( 𝐴 ≼* 𝐵 ↔ 𝐴 ≼ 𝐵 ) ) |