Step |
Hyp |
Ref |
Expression |
1 |
|
eldif |
⊢ ( 𝑏 ∈ ( On ∖ ω ) ↔ ( 𝑏 ∈ On ∧ ¬ 𝑏 ∈ ω ) ) |
2 |
|
enfi |
⊢ ( 𝐴 ≈ 𝑏 → ( 𝐴 ∈ Fin ↔ 𝑏 ∈ Fin ) ) |
3 |
|
onfin |
⊢ ( 𝑏 ∈ On → ( 𝑏 ∈ Fin ↔ 𝑏 ∈ ω ) ) |
4 |
2 3
|
sylan9bbr |
⊢ ( ( 𝑏 ∈ On ∧ 𝐴 ≈ 𝑏 ) → ( 𝐴 ∈ Fin ↔ 𝑏 ∈ ω ) ) |
5 |
4
|
biimpd |
⊢ ( ( 𝑏 ∈ On ∧ 𝐴 ≈ 𝑏 ) → ( 𝐴 ∈ Fin → 𝑏 ∈ ω ) ) |
6 |
5
|
con3d |
⊢ ( ( 𝑏 ∈ On ∧ 𝐴 ≈ 𝑏 ) → ( ¬ 𝑏 ∈ ω → ¬ 𝐴 ∈ Fin ) ) |
7 |
6
|
impancom |
⊢ ( ( 𝑏 ∈ On ∧ ¬ 𝑏 ∈ ω ) → ( 𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin ) ) |
8 |
1 7
|
sylbi |
⊢ ( 𝑏 ∈ ( On ∖ ω ) → ( 𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin ) ) |
9 |
8
|
rexlimiv |
⊢ ( ∃ 𝑏 ∈ ( On ∖ ω ) 𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin ) |
10 |
9
|
con2i |
⊢ ( 𝐴 ∈ Fin → ¬ ∃ 𝑏 ∈ ( On ∖ ω ) 𝐴 ≈ 𝑏 ) |
11 |
|
isfin7 |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∈ FinVII ↔ ¬ ∃ 𝑏 ∈ ( On ∖ ω ) 𝐴 ≈ 𝑏 ) ) |
12 |
10 11
|
mpbird |
⊢ ( 𝐴 ∈ Fin → 𝐴 ∈ FinVII ) |