Step |
Hyp |
Ref |
Expression |
1 |
|
nn0re |
⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ ) |
2 |
|
ltpnf |
⊢ ( 𝐵 ∈ ℝ → 𝐵 < +∞ ) |
3 |
|
rexr |
⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* ) |
4 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
5 |
|
xrltnle |
⊢ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 𝐵 < +∞ ↔ ¬ +∞ ≤ 𝐵 ) ) |
6 |
3 4 5
|
sylancl |
⊢ ( 𝐵 ∈ ℝ → ( 𝐵 < +∞ ↔ ¬ +∞ ≤ 𝐵 ) ) |
7 |
2 6
|
mpbid |
⊢ ( 𝐵 ∈ ℝ → ¬ +∞ ≤ 𝐵 ) |
8 |
1 7
|
syl |
⊢ ( 𝐵 ∈ ℕ0 → ¬ +∞ ≤ 𝐵 ) |
9 |
|
hashinf |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ( ♯ ‘ 𝐴 ) = +∞ ) |
10 |
9
|
breq1d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ( ( ♯ ‘ 𝐴 ) ≤ 𝐵 ↔ +∞ ≤ 𝐵 ) ) |
11 |
10
|
notbid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ( ¬ ( ♯ ‘ 𝐴 ) ≤ 𝐵 ↔ ¬ +∞ ≤ 𝐵 ) ) |
12 |
8 11
|
syl5ibrcom |
⊢ ( 𝐵 ∈ ℕ0 → ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ¬ ( ♯ ‘ 𝐴 ) ≤ 𝐵 ) ) |
13 |
12
|
expdimp |
⊢ ( ( 𝐵 ∈ ℕ0 ∧ 𝐴 ∈ 𝑉 ) → ( ¬ 𝐴 ∈ Fin → ¬ ( ♯ ‘ 𝐴 ) ≤ 𝐵 ) ) |
14 |
13
|
ancoms |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0 ) → ( ¬ 𝐴 ∈ Fin → ¬ ( ♯ ‘ 𝐴 ) ≤ 𝐵 ) ) |
15 |
14
|
con4d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0 ) → ( ( ♯ ‘ 𝐴 ) ≤ 𝐵 → 𝐴 ∈ Fin ) ) |
16 |
15
|
3impia |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0 ∧ ( ♯ ‘ 𝐴 ) ≤ 𝐵 ) → 𝐴 ∈ Fin ) |