Step |
Hyp |
Ref |
Expression |
1 |
|
id |
⊢ ( ( ♯ ‘ 𝑉 ) = 1 → ( ♯ ‘ 𝑉 ) = 1 ) |
2 |
|
hash1 |
⊢ ( ♯ ‘ 1o ) = 1 |
3 |
1 2
|
eqtr4di |
⊢ ( ( ♯ ‘ 𝑉 ) = 1 → ( ♯ ‘ 𝑉 ) = ( ♯ ‘ 1o ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝑉 ∈ Fin ∧ ( ♯ ‘ 𝑉 ) = 1 ) → ( ♯ ‘ 𝑉 ) = ( ♯ ‘ 1o ) ) |
5 |
|
1onn |
⊢ 1o ∈ ω |
6 |
|
nnfi |
⊢ ( 1o ∈ ω → 1o ∈ Fin ) |
7 |
5 6
|
mp1i |
⊢ ( ( ♯ ‘ 𝑉 ) = 1 → 1o ∈ Fin ) |
8 |
|
hashen |
⊢ ( ( 𝑉 ∈ Fin ∧ 1o ∈ Fin ) → ( ( ♯ ‘ 𝑉 ) = ( ♯ ‘ 1o ) ↔ 𝑉 ≈ 1o ) ) |
9 |
7 8
|
sylan2 |
⊢ ( ( 𝑉 ∈ Fin ∧ ( ♯ ‘ 𝑉 ) = 1 ) → ( ( ♯ ‘ 𝑉 ) = ( ♯ ‘ 1o ) ↔ 𝑉 ≈ 1o ) ) |
10 |
4 9
|
mpbid |
⊢ ( ( 𝑉 ∈ Fin ∧ ( ♯ ‘ 𝑉 ) = 1 ) → 𝑉 ≈ 1o ) |
11 |
|
en1 |
⊢ ( 𝑉 ≈ 1o ↔ ∃ 𝑎 𝑉 = { 𝑎 } ) |
12 |
10 11
|
sylib |
⊢ ( ( 𝑉 ∈ Fin ∧ ( ♯ ‘ 𝑉 ) = 1 ) → ∃ 𝑎 𝑉 = { 𝑎 } ) |
13 |
12
|
ex |
⊢ ( 𝑉 ∈ Fin → ( ( ♯ ‘ 𝑉 ) = 1 → ∃ 𝑎 𝑉 = { 𝑎 } ) ) |
14 |
13
|
a1d |
⊢ ( 𝑉 ∈ Fin → ( 𝑉 ∈ 𝑊 → ( ( ♯ ‘ 𝑉 ) = 1 → ∃ 𝑎 𝑉 = { 𝑎 } ) ) ) |
15 |
|
hashinf |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ ¬ 𝑉 ∈ Fin ) → ( ♯ ‘ 𝑉 ) = +∞ ) |
16 |
|
eqeq1 |
⊢ ( ( ♯ ‘ 𝑉 ) = +∞ → ( ( ♯ ‘ 𝑉 ) = 1 ↔ +∞ = 1 ) ) |
17 |
|
1re |
⊢ 1 ∈ ℝ |
18 |
|
renepnf |
⊢ ( 1 ∈ ℝ → 1 ≠ +∞ ) |
19 |
|
df-ne |
⊢ ( 1 ≠ +∞ ↔ ¬ 1 = +∞ ) |
20 |
|
pm2.21 |
⊢ ( ¬ 1 = +∞ → ( 1 = +∞ → ∃ 𝑎 𝑉 = { 𝑎 } ) ) |
21 |
19 20
|
sylbi |
⊢ ( 1 ≠ +∞ → ( 1 = +∞ → ∃ 𝑎 𝑉 = { 𝑎 } ) ) |
22 |
17 18 21
|
mp2b |
⊢ ( 1 = +∞ → ∃ 𝑎 𝑉 = { 𝑎 } ) |
23 |
22
|
eqcoms |
⊢ ( +∞ = 1 → ∃ 𝑎 𝑉 = { 𝑎 } ) |
24 |
16 23
|
syl6bi |
⊢ ( ( ♯ ‘ 𝑉 ) = +∞ → ( ( ♯ ‘ 𝑉 ) = 1 → ∃ 𝑎 𝑉 = { 𝑎 } ) ) |
25 |
15 24
|
syl |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ ¬ 𝑉 ∈ Fin ) → ( ( ♯ ‘ 𝑉 ) = 1 → ∃ 𝑎 𝑉 = { 𝑎 } ) ) |
26 |
25
|
expcom |
⊢ ( ¬ 𝑉 ∈ Fin → ( 𝑉 ∈ 𝑊 → ( ( ♯ ‘ 𝑉 ) = 1 → ∃ 𝑎 𝑉 = { 𝑎 } ) ) ) |
27 |
14 26
|
pm2.61i |
⊢ ( 𝑉 ∈ 𝑊 → ( ( ♯ ‘ 𝑉 ) = 1 → ∃ 𝑎 𝑉 = { 𝑎 } ) ) |
28 |
|
fveq2 |
⊢ ( 𝑉 = { 𝑎 } → ( ♯ ‘ 𝑉 ) = ( ♯ ‘ { 𝑎 } ) ) |
29 |
|
hashsng |
⊢ ( 𝑎 ∈ V → ( ♯ ‘ { 𝑎 } ) = 1 ) |
30 |
29
|
elv |
⊢ ( ♯ ‘ { 𝑎 } ) = 1 |
31 |
28 30
|
eqtrdi |
⊢ ( 𝑉 = { 𝑎 } → ( ♯ ‘ 𝑉 ) = 1 ) |
32 |
31
|
exlimiv |
⊢ ( ∃ 𝑎 𝑉 = { 𝑎 } → ( ♯ ‘ 𝑉 ) = 1 ) |
33 |
27 32
|
impbid1 |
⊢ ( 𝑉 ∈ 𝑊 → ( ( ♯ ‘ 𝑉 ) = 1 ↔ ∃ 𝑎 𝑉 = { 𝑎 } ) ) |