Step |
Hyp |
Ref |
Expression |
1 |
|
alephon |
⊢ ( ℵ ‘ 𝐴 ) ∈ On |
2 |
|
id |
⊢ ( ( card ‘ 𝐵 ) = 𝐵 → ( card ‘ 𝐵 ) = 𝐵 ) |
3 |
|
cardon |
⊢ ( card ‘ 𝐵 ) ∈ On |
4 |
2 3
|
eqeltrrdi |
⊢ ( ( card ‘ 𝐵 ) = 𝐵 → 𝐵 ∈ On ) |
5 |
|
onenon |
⊢ ( 𝐵 ∈ On → 𝐵 ∈ dom card ) |
6 |
4 5
|
syl |
⊢ ( ( card ‘ 𝐵 ) = 𝐵 → 𝐵 ∈ dom card ) |
7 |
|
cardsdomel |
⊢ ( ( ( ℵ ‘ 𝐴 ) ∈ On ∧ 𝐵 ∈ dom card ) → ( ( ℵ ‘ 𝐴 ) ≺ 𝐵 ↔ ( ℵ ‘ 𝐴 ) ∈ ( card ‘ 𝐵 ) ) ) |
8 |
1 6 7
|
sylancr |
⊢ ( ( card ‘ 𝐵 ) = 𝐵 → ( ( ℵ ‘ 𝐴 ) ≺ 𝐵 ↔ ( ℵ ‘ 𝐴 ) ∈ ( card ‘ 𝐵 ) ) ) |
9 |
|
eleq2 |
⊢ ( ( card ‘ 𝐵 ) = 𝐵 → ( ( ℵ ‘ 𝐴 ) ∈ ( card ‘ 𝐵 ) ↔ ( ℵ ‘ 𝐴 ) ∈ 𝐵 ) ) |
10 |
8 9
|
bitrd |
⊢ ( ( card ‘ 𝐵 ) = 𝐵 → ( ( ℵ ‘ 𝐴 ) ≺ 𝐵 ↔ ( ℵ ‘ 𝐴 ) ∈ 𝐵 ) ) |
11 |
10
|
adantl |
⊢ ( ( 𝐴 ∈ On ∧ ( card ‘ 𝐵 ) = 𝐵 ) → ( ( ℵ ‘ 𝐴 ) ≺ 𝐵 ↔ ( ℵ ‘ 𝐴 ) ∈ 𝐵 ) ) |
12 |
|
alephsuc |
⊢ ( 𝐴 ∈ On → ( ℵ ‘ suc 𝐴 ) = ( har ‘ ( ℵ ‘ 𝐴 ) ) ) |
13 |
|
onenon |
⊢ ( ( ℵ ‘ 𝐴 ) ∈ On → ( ℵ ‘ 𝐴 ) ∈ dom card ) |
14 |
|
harval2 |
⊢ ( ( ℵ ‘ 𝐴 ) ∈ dom card → ( har ‘ ( ℵ ‘ 𝐴 ) ) = ∩ { 𝑥 ∈ On ∣ ( ℵ ‘ 𝐴 ) ≺ 𝑥 } ) |
15 |
1 13 14
|
mp2b |
⊢ ( har ‘ ( ℵ ‘ 𝐴 ) ) = ∩ { 𝑥 ∈ On ∣ ( ℵ ‘ 𝐴 ) ≺ 𝑥 } |
16 |
12 15
|
eqtrdi |
⊢ ( 𝐴 ∈ On → ( ℵ ‘ suc 𝐴 ) = ∩ { 𝑥 ∈ On ∣ ( ℵ ‘ 𝐴 ) ≺ 𝑥 } ) |
17 |
16
|
eleq2d |
⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ↔ 𝐵 ∈ ∩ { 𝑥 ∈ On ∣ ( ℵ ‘ 𝐴 ) ≺ 𝑥 } ) ) |
18 |
17
|
biimpd |
⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) → 𝐵 ∈ ∩ { 𝑥 ∈ On ∣ ( ℵ ‘ 𝐴 ) ≺ 𝑥 } ) ) |
19 |
|
breq2 |
⊢ ( 𝑥 = 𝐵 → ( ( ℵ ‘ 𝐴 ) ≺ 𝑥 ↔ ( ℵ ‘ 𝐴 ) ≺ 𝐵 ) ) |
20 |
19
|
onnminsb |
⊢ ( 𝐵 ∈ On → ( 𝐵 ∈ ∩ { 𝑥 ∈ On ∣ ( ℵ ‘ 𝐴 ) ≺ 𝑥 } → ¬ ( ℵ ‘ 𝐴 ) ≺ 𝐵 ) ) |
21 |
18 20
|
sylan9 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) → ¬ ( ℵ ‘ 𝐴 ) ≺ 𝐵 ) ) |
22 |
21
|
con2d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( ℵ ‘ 𝐴 ) ≺ 𝐵 → ¬ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) ) |
23 |
4 22
|
sylan2 |
⊢ ( ( 𝐴 ∈ On ∧ ( card ‘ 𝐵 ) = 𝐵 ) → ( ( ℵ ‘ 𝐴 ) ≺ 𝐵 → ¬ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) ) |
24 |
11 23
|
sylbird |
⊢ ( ( 𝐴 ∈ On ∧ ( card ‘ 𝐵 ) = 𝐵 ) → ( ( ℵ ‘ 𝐴 ) ∈ 𝐵 → ¬ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) ) |
25 |
|
imnan |
⊢ ( ( ( ℵ ‘ 𝐴 ) ∈ 𝐵 → ¬ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) ↔ ¬ ( ( ℵ ‘ 𝐴 ) ∈ 𝐵 ∧ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) ) |
26 |
24 25
|
sylib |
⊢ ( ( 𝐴 ∈ On ∧ ( card ‘ 𝐵 ) = 𝐵 ) → ¬ ( ( ℵ ‘ 𝐴 ) ∈ 𝐵 ∧ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) ) |
27 |
26
|
ex |
⊢ ( 𝐴 ∈ On → ( ( card ‘ 𝐵 ) = 𝐵 → ¬ ( ( ℵ ‘ 𝐴 ) ∈ 𝐵 ∧ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) ) ) |
28 |
|
n0i |
⊢ ( 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) → ¬ ( ℵ ‘ suc 𝐴 ) = ∅ ) |
29 |
|
alephfnon |
⊢ ℵ Fn On |
30 |
29
|
fndmi |
⊢ dom ℵ = On |
31 |
30
|
eleq2i |
⊢ ( suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On ) |
32 |
|
ndmfv |
⊢ ( ¬ suc 𝐴 ∈ dom ℵ → ( ℵ ‘ suc 𝐴 ) = ∅ ) |
33 |
31 32
|
sylnbir |
⊢ ( ¬ suc 𝐴 ∈ On → ( ℵ ‘ suc 𝐴 ) = ∅ ) |
34 |
28 33
|
nsyl2 |
⊢ ( 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) → suc 𝐴 ∈ On ) |
35 |
|
sucelon |
⊢ ( 𝐴 ∈ On ↔ suc 𝐴 ∈ On ) |
36 |
34 35
|
sylibr |
⊢ ( 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) → 𝐴 ∈ On ) |
37 |
36
|
adantl |
⊢ ( ( ( ℵ ‘ 𝐴 ) ∈ 𝐵 ∧ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) → 𝐴 ∈ On ) |
38 |
37
|
con3i |
⊢ ( ¬ 𝐴 ∈ On → ¬ ( ( ℵ ‘ 𝐴 ) ∈ 𝐵 ∧ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) ) |
39 |
38
|
a1d |
⊢ ( ¬ 𝐴 ∈ On → ( ( card ‘ 𝐵 ) = 𝐵 → ¬ ( ( ℵ ‘ 𝐴 ) ∈ 𝐵 ∧ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) ) ) |
40 |
27 39
|
pm2.61i |
⊢ ( ( card ‘ 𝐵 ) = 𝐵 → ¬ ( ( ℵ ‘ 𝐴 ) ∈ 𝐵 ∧ 𝐵 ∈ ( ℵ ‘ suc 𝐴 ) ) ) |