Step |
Hyp |
Ref |
Expression |
1 |
|
rankr1ai |
⊢ ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) → ( rank ‘ 𝐴 ) ∈ 𝐵 ) |
2 |
|
r1funlim |
⊢ ( Fun 𝑅1 ∧ Lim dom 𝑅1 ) |
3 |
2
|
simpri |
⊢ Lim dom 𝑅1 |
4 |
|
limord |
⊢ ( Lim dom 𝑅1 → Ord dom 𝑅1 ) |
5 |
3 4
|
ax-mp |
⊢ Ord dom 𝑅1 |
6 |
|
ordelord |
⊢ ( ( Ord dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1 ) → Ord 𝐵 ) |
7 |
5 6
|
mpan |
⊢ ( 𝐵 ∈ dom 𝑅1 → Ord 𝐵 ) |
8 |
7
|
adantl |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → Ord 𝐵 ) |
9 |
|
ordsucss |
⊢ ( Ord 𝐵 → ( ( rank ‘ 𝐴 ) ∈ 𝐵 → suc ( rank ‘ 𝐴 ) ⊆ 𝐵 ) ) |
10 |
8 9
|
syl |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( ( rank ‘ 𝐴 ) ∈ 𝐵 → suc ( rank ‘ 𝐴 ) ⊆ 𝐵 ) ) |
11 |
|
rankidb |
⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) ) |
12 |
|
elfvdm |
⊢ ( 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) → suc ( rank ‘ 𝐴 ) ∈ dom 𝑅1 ) |
13 |
11 12
|
syl |
⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → suc ( rank ‘ 𝐴 ) ∈ dom 𝑅1 ) |
14 |
|
r1ord3g |
⊢ ( ( suc ( rank ‘ 𝐴 ) ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1 ) → ( suc ( rank ‘ 𝐴 ) ⊆ 𝐵 → ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) ⊆ ( 𝑅1 ‘ 𝐵 ) ) ) |
15 |
13 14
|
sylan |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( suc ( rank ‘ 𝐴 ) ⊆ 𝐵 → ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) ⊆ ( 𝑅1 ‘ 𝐵 ) ) ) |
16 |
11
|
adantr |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) ) |
17 |
|
ssel |
⊢ ( ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) ⊆ ( 𝑅1 ‘ 𝐵 ) → ( 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) → 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ) ) |
18 |
16 17
|
syl5com |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) ⊆ ( 𝑅1 ‘ 𝐵 ) → 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ) ) |
19 |
10 15 18
|
3syld |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( ( rank ‘ 𝐴 ) ∈ 𝐵 → 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ) ) |
20 |
1 19
|
impbid2 |
⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ ( rank ‘ 𝐴 ) ∈ 𝐵 ) ) |