Step |
Hyp |
Ref |
Expression |
1 |
|
1onn |
⊢ 1o ∈ ω |
2 |
|
ssid |
⊢ 1o ⊆ 1o |
3 |
|
ssnnfi |
⊢ ( ( 1o ∈ ω ∧ 1o ⊆ 1o ) → 1o ∈ Fin ) |
4 |
1 2 3
|
mp2an |
⊢ 1o ∈ Fin |
5 |
|
enfii |
⊢ ( ( 1o ∈ Fin ∧ 𝐵 ≈ 1o ) → 𝐵 ∈ Fin ) |
6 |
4 5
|
mpan |
⊢ ( 𝐵 ≈ 1o → 𝐵 ∈ Fin ) |
7 |
6
|
adantl |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o ) → 𝐵 ∈ Fin ) |
8 |
|
snssi |
⊢ ( 𝐴 ∈ 𝐵 → { 𝐴 } ⊆ 𝐵 ) |
9 |
8
|
adantr |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o ) → { 𝐴 } ⊆ 𝐵 ) |
10 |
|
ensn1g |
⊢ ( 𝐴 ∈ 𝐵 → { 𝐴 } ≈ 1o ) |
11 |
|
ensym |
⊢ ( 𝐵 ≈ 1o → 1o ≈ 𝐵 ) |
12 |
|
entr |
⊢ ( ( { 𝐴 } ≈ 1o ∧ 1o ≈ 𝐵 ) → { 𝐴 } ≈ 𝐵 ) |
13 |
10 11 12
|
syl2an |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o ) → { 𝐴 } ≈ 𝐵 ) |
14 |
|
fisseneq |
⊢ ( ( 𝐵 ∈ Fin ∧ { 𝐴 } ⊆ 𝐵 ∧ { 𝐴 } ≈ 𝐵 ) → { 𝐴 } = 𝐵 ) |
15 |
7 9 13 14
|
syl3anc |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o ) → { 𝐴 } = 𝐵 ) |
16 |
15
|
eqcomd |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o ) → 𝐵 = { 𝐴 } ) |