Step |
Hyp |
Ref |
Expression |
1 |
|
1onn |
⊢ 1o ∈ ω |
2 |
1
|
a1i |
⊢ ( ¬ 𝐴 ∈ Fin → 1o ∈ ω ) |
3 |
|
djudoml |
⊢ ( ( 𝐴 ∈ GCH ∧ 1o ∈ ω ) → 𝐴 ≼ ( 𝐴 ⊔ 1o ) ) |
4 |
2 3
|
sylan2 |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → 𝐴 ≼ ( 𝐴 ⊔ 1o ) ) |
5 |
|
simpr |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → ¬ 𝐴 ∈ Fin ) |
6 |
|
nnfi |
⊢ ( 1o ∈ ω → 1o ∈ Fin ) |
7 |
1 6
|
mp1i |
⊢ ( ¬ 𝐴 ∈ Fin → 1o ∈ Fin ) |
8 |
|
fidomtri2 |
⊢ ( ( 𝐴 ∈ GCH ∧ 1o ∈ Fin ) → ( 𝐴 ≼ 1o ↔ ¬ 1o ≺ 𝐴 ) ) |
9 |
7 8
|
sylan2 |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → ( 𝐴 ≼ 1o ↔ ¬ 1o ≺ 𝐴 ) ) |
10 |
1 6
|
mp1i |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → 1o ∈ Fin ) |
11 |
|
domfi |
⊢ ( ( 1o ∈ Fin ∧ 𝐴 ≼ 1o ) → 𝐴 ∈ Fin ) |
12 |
11
|
ex |
⊢ ( 1o ∈ Fin → ( 𝐴 ≼ 1o → 𝐴 ∈ Fin ) ) |
13 |
10 12
|
syl |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → ( 𝐴 ≼ 1o → 𝐴 ∈ Fin ) ) |
14 |
9 13
|
sylbird |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → ( ¬ 1o ≺ 𝐴 → 𝐴 ∈ Fin ) ) |
15 |
5 14
|
mt3d |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → 1o ≺ 𝐴 ) |
16 |
|
canthp1 |
⊢ ( 1o ≺ 𝐴 → ( 𝐴 ⊔ 1o ) ≺ 𝒫 𝐴 ) |
17 |
15 16
|
syl |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → ( 𝐴 ⊔ 1o ) ≺ 𝒫 𝐴 ) |
18 |
4 17
|
jca |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → ( 𝐴 ≼ ( 𝐴 ⊔ 1o ) ∧ ( 𝐴 ⊔ 1o ) ≺ 𝒫 𝐴 ) ) |
19 |
|
gchen1 |
⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ ( 𝐴 ⊔ 1o ) ∧ ( 𝐴 ⊔ 1o ) ≺ 𝒫 𝐴 ) ) → 𝐴 ≈ ( 𝐴 ⊔ 1o ) ) |
20 |
18 19
|
mpdan |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → 𝐴 ≈ ( 𝐴 ⊔ 1o ) ) |
21 |
20
|
ensymd |
⊢ ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) → ( 𝐴 ⊔ 1o ) ≈ 𝐴 ) |