Step |
Hyp |
Ref |
Expression |
1 |
|
scutbdaybnd2 |
⊢ ( 𝐴 <<s 𝐵 → ( bday ‘ ( 𝐴 |s 𝐵 ) ) ⊆ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) |
2 |
1
|
adantr |
⊢ ( ( 𝐴 <<s 𝐵 ∧ Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ) → ( bday ‘ ( 𝐴 |s 𝐵 ) ) ⊆ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) |
3 |
|
bdayfun |
⊢ Fun bday |
4 |
|
ssltex1 |
⊢ ( 𝐴 <<s 𝐵 → 𝐴 ∈ V ) |
5 |
|
ssltex2 |
⊢ ( 𝐴 <<s 𝐵 → 𝐵 ∈ V ) |
6 |
|
unexg |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ∪ 𝐵 ) ∈ V ) |
7 |
4 5 6
|
syl2anc |
⊢ ( 𝐴 <<s 𝐵 → ( 𝐴 ∪ 𝐵 ) ∈ V ) |
8 |
|
funimaexg |
⊢ ( ( Fun bday ∧ ( 𝐴 ∪ 𝐵 ) ∈ V ) → ( bday “ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) |
9 |
3 7 8
|
sylancr |
⊢ ( 𝐴 <<s 𝐵 → ( bday “ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) |
10 |
9
|
uniexd |
⊢ ( 𝐴 <<s 𝐵 → ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) |
11 |
10
|
adantr |
⊢ ( ( 𝐴 <<s 𝐵 ∧ Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ) → ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) |
12 |
|
nlimsucg |
⊢ ( ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ∈ V → ¬ Lim suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) |
13 |
11 12
|
syl |
⊢ ( ( 𝐴 <<s 𝐵 ∧ Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ) → ¬ Lim suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) |
14 |
|
limeq |
⊢ ( ( bday ‘ ( 𝐴 |s 𝐵 ) ) = suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) → ( Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ↔ Lim suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) ) |
15 |
14
|
biimpcd |
⊢ ( Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) → ( ( bday ‘ ( 𝐴 |s 𝐵 ) ) = suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) → Lim suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) ) |
16 |
15
|
adantl |
⊢ ( ( 𝐴 <<s 𝐵 ∧ Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ) → ( ( bday ‘ ( 𝐴 |s 𝐵 ) ) = suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) → Lim suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) ) |
17 |
13 16
|
mtod |
⊢ ( ( 𝐴 <<s 𝐵 ∧ Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ) → ¬ ( bday ‘ ( 𝐴 |s 𝐵 ) ) = suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) |
18 |
17
|
neqned |
⊢ ( ( 𝐴 <<s 𝐵 ∧ Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ) → ( bday ‘ ( 𝐴 |s 𝐵 ) ) ≠ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) |
19 |
|
bdayelon |
⊢ ( bday ‘ ( 𝐴 |s 𝐵 ) ) ∈ On |
20 |
19
|
onordi |
⊢ Ord ( bday ‘ ( 𝐴 |s 𝐵 ) ) |
21 |
|
imassrn |
⊢ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ⊆ ran bday |
22 |
|
bdayrn |
⊢ ran bday = On |
23 |
21 22
|
sseqtri |
⊢ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ⊆ On |
24 |
|
ssorduni |
⊢ ( ( bday “ ( 𝐴 ∪ 𝐵 ) ) ⊆ On → Ord ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) |
25 |
23 24
|
ax-mp |
⊢ Ord ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) |
26 |
|
ordsuc |
⊢ ( Ord ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ↔ Ord suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) |
27 |
25 26
|
mpbi |
⊢ Ord suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) |
28 |
|
ordelssne |
⊢ ( ( Ord ( bday ‘ ( 𝐴 |s 𝐵 ) ) ∧ Ord suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) → ( ( bday ‘ ( 𝐴 |s 𝐵 ) ) ∈ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ↔ ( ( bday ‘ ( 𝐴 |s 𝐵 ) ) ⊆ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ∧ ( bday ‘ ( 𝐴 |s 𝐵 ) ) ≠ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) ) ) |
29 |
20 27 28
|
mp2an |
⊢ ( ( bday ‘ ( 𝐴 |s 𝐵 ) ) ∈ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ↔ ( ( bday ‘ ( 𝐴 |s 𝐵 ) ) ⊆ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ∧ ( bday ‘ ( 𝐴 |s 𝐵 ) ) ≠ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) ) |
30 |
2 18 29
|
sylanbrc |
⊢ ( ( 𝐴 <<s 𝐵 ∧ Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ) → ( bday ‘ ( 𝐴 |s 𝐵 ) ) ∈ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) |
31 |
19
|
a1i |
⊢ ( ( 𝐴 <<s 𝐵 ∧ Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ) → ( bday ‘ ( 𝐴 |s 𝐵 ) ) ∈ On ) |
32 |
|
ordsssuc |
⊢ ( ( ( bday ‘ ( 𝐴 |s 𝐵 ) ) ∈ On ∧ Ord ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) → ( ( bday ‘ ( 𝐴 |s 𝐵 ) ) ⊆ ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ↔ ( bday ‘ ( 𝐴 |s 𝐵 ) ) ∈ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) ) |
33 |
31 25 32
|
sylancl |
⊢ ( ( 𝐴 <<s 𝐵 ∧ Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ) → ( ( bday ‘ ( 𝐴 |s 𝐵 ) ) ⊆ ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ↔ ( bday ‘ ( 𝐴 |s 𝐵 ) ) ∈ suc ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) ) |
34 |
30 33
|
mpbird |
⊢ ( ( 𝐴 <<s 𝐵 ∧ Lim ( bday ‘ ( 𝐴 |s 𝐵 ) ) ) → ( bday ‘ ( 𝐴 |s 𝐵 ) ) ⊆ ∪ ( bday “ ( 𝐴 ∪ 𝐵 ) ) ) |