Step |
Hyp |
Ref |
Expression |
1 |
|
onsseleq |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) ) |
2 |
|
alephord |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ∈ 𝐵 ↔ ( ℵ ‘ 𝐴 ) ≺ ( ℵ ‘ 𝐵 ) ) ) |
3 |
|
sdomdom |
⊢ ( ( ℵ ‘ 𝐴 ) ≺ ( ℵ ‘ 𝐵 ) → ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) |
4 |
2 3
|
syl6bi |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) ) |
5 |
|
fvex |
⊢ ( ℵ ‘ 𝐴 ) ∈ V |
6 |
|
fveq2 |
⊢ ( 𝐴 = 𝐵 → ( ℵ ‘ 𝐴 ) = ( ℵ ‘ 𝐵 ) ) |
7 |
|
eqeng |
⊢ ( ( ℵ ‘ 𝐴 ) ∈ V → ( ( ℵ ‘ 𝐴 ) = ( ℵ ‘ 𝐵 ) → ( ℵ ‘ 𝐴 ) ≈ ( ℵ ‘ 𝐵 ) ) ) |
8 |
5 6 7
|
mpsyl |
⊢ ( 𝐴 = 𝐵 → ( ℵ ‘ 𝐴 ) ≈ ( ℵ ‘ 𝐵 ) ) |
9 |
8
|
a1i |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 = 𝐵 → ( ℵ ‘ 𝐴 ) ≈ ( ℵ ‘ 𝐵 ) ) ) |
10 |
|
endom |
⊢ ( ( ℵ ‘ 𝐴 ) ≈ ( ℵ ‘ 𝐵 ) → ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) |
11 |
9 10
|
syl6 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 = 𝐵 → ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) ) |
12 |
4 11
|
jaod |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) → ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) ) |
13 |
1 12
|
sylbid |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) ) |
14 |
|
eloni |
⊢ ( 𝐵 ∈ On → Ord 𝐵 ) |
15 |
|
eloni |
⊢ ( 𝐴 ∈ On → Ord 𝐴 ) |
16 |
|
ordtri2or |
⊢ ( ( Ord 𝐵 ∧ Ord 𝐴 ) → ( 𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵 ) ) |
17 |
14 15 16
|
syl2anr |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵 ) ) |
18 |
17
|
ord |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ¬ 𝐵 ∈ 𝐴 → 𝐴 ⊆ 𝐵 ) ) |
19 |
18
|
con1d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ¬ 𝐴 ⊆ 𝐵 → 𝐵 ∈ 𝐴 ) ) |
20 |
|
alephord |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( 𝐵 ∈ 𝐴 ↔ ( ℵ ‘ 𝐵 ) ≺ ( ℵ ‘ 𝐴 ) ) ) |
21 |
20
|
ancoms |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 ∈ 𝐴 ↔ ( ℵ ‘ 𝐵 ) ≺ ( ℵ ‘ 𝐴 ) ) ) |
22 |
|
sdomnen |
⊢ ( ( ℵ ‘ 𝐵 ) ≺ ( ℵ ‘ 𝐴 ) → ¬ ( ℵ ‘ 𝐵 ) ≈ ( ℵ ‘ 𝐴 ) ) |
23 |
|
sdomdom |
⊢ ( ( ℵ ‘ 𝐵 ) ≺ ( ℵ ‘ 𝐴 ) → ( ℵ ‘ 𝐵 ) ≼ ( ℵ ‘ 𝐴 ) ) |
24 |
|
sbth |
⊢ ( ( ( ℵ ‘ 𝐵 ) ≼ ( ℵ ‘ 𝐴 ) ∧ ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) → ( ℵ ‘ 𝐵 ) ≈ ( ℵ ‘ 𝐴 ) ) |
25 |
24
|
ex |
⊢ ( ( ℵ ‘ 𝐵 ) ≼ ( ℵ ‘ 𝐴 ) → ( ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) → ( ℵ ‘ 𝐵 ) ≈ ( ℵ ‘ 𝐴 ) ) ) |
26 |
23 25
|
syl |
⊢ ( ( ℵ ‘ 𝐵 ) ≺ ( ℵ ‘ 𝐴 ) → ( ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) → ( ℵ ‘ 𝐵 ) ≈ ( ℵ ‘ 𝐴 ) ) ) |
27 |
22 26
|
mtod |
⊢ ( ( ℵ ‘ 𝐵 ) ≺ ( ℵ ‘ 𝐴 ) → ¬ ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) |
28 |
21 27
|
syl6bi |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 ∈ 𝐴 → ¬ ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) ) |
29 |
19 28
|
syld |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ¬ 𝐴 ⊆ 𝐵 → ¬ ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) ) |
30 |
13 29
|
impcon4bid |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( ℵ ‘ 𝐴 ) ≼ ( ℵ ‘ 𝐵 ) ) ) |