Step |
Hyp |
Ref |
Expression |
1 |
|
r1tr |
⊢ Tr ( 𝑅1 ‘ 𝐴 ) |
2 |
1
|
a1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) → Tr ( 𝑅1 ‘ 𝐴 ) ) |
3 |
|
limelon |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) → 𝐴 ∈ On ) |
4 |
|
r1fnon |
⊢ 𝑅1 Fn On |
5 |
4
|
fndmi |
⊢ dom 𝑅1 = On |
6 |
3 5
|
eleqtrrdi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) → 𝐴 ∈ dom 𝑅1 ) |
7 |
|
onssr1 |
⊢ ( 𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ ( 𝑅1 ‘ 𝐴 ) ) |
8 |
6 7
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) → 𝐴 ⊆ ( 𝑅1 ‘ 𝐴 ) ) |
9 |
|
0ellim |
⊢ ( Lim 𝐴 → ∅ ∈ 𝐴 ) |
10 |
9
|
adantl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) → ∅ ∈ 𝐴 ) |
11 |
8 10
|
sseldd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) → ∅ ∈ ( 𝑅1 ‘ 𝐴 ) ) |
12 |
11
|
ne0d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) → ( 𝑅1 ‘ 𝐴 ) ≠ ∅ ) |
13 |
|
rankuni |
⊢ ( rank ‘ ∪ 𝑥 ) = ∪ ( rank ‘ 𝑥 ) |
14 |
|
rankon |
⊢ ( rank ‘ 𝑥 ) ∈ On |
15 |
|
eloni |
⊢ ( ( rank ‘ 𝑥 ) ∈ On → Ord ( rank ‘ 𝑥 ) ) |
16 |
|
orduniss |
⊢ ( Ord ( rank ‘ 𝑥 ) → ∪ ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑥 ) ) |
17 |
14 15 16
|
mp2b |
⊢ ∪ ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑥 ) |
18 |
17
|
a1i |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ∪ ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑥 ) ) |
19 |
|
rankr1ai |
⊢ ( 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) → ( rank ‘ 𝑥 ) ∈ 𝐴 ) |
20 |
19
|
adantl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( rank ‘ 𝑥 ) ∈ 𝐴 ) |
21 |
|
onuni |
⊢ ( ( rank ‘ 𝑥 ) ∈ On → ∪ ( rank ‘ 𝑥 ) ∈ On ) |
22 |
14 21
|
ax-mp |
⊢ ∪ ( rank ‘ 𝑥 ) ∈ On |
23 |
3
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → 𝐴 ∈ On ) |
24 |
|
ontr2 |
⊢ ( ( ∪ ( rank ‘ 𝑥 ) ∈ On ∧ 𝐴 ∈ On ) → ( ( ∪ ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑥 ) ∧ ( rank ‘ 𝑥 ) ∈ 𝐴 ) → ∪ ( rank ‘ 𝑥 ) ∈ 𝐴 ) ) |
25 |
22 23 24
|
sylancr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( ( ∪ ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑥 ) ∧ ( rank ‘ 𝑥 ) ∈ 𝐴 ) → ∪ ( rank ‘ 𝑥 ) ∈ 𝐴 ) ) |
26 |
18 20 25
|
mp2and |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ∪ ( rank ‘ 𝑥 ) ∈ 𝐴 ) |
27 |
13 26
|
eqeltrid |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( rank ‘ ∪ 𝑥 ) ∈ 𝐴 ) |
28 |
|
r1elwf |
⊢ ( 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) → 𝑥 ∈ ∪ ( 𝑅1 “ On ) ) |
29 |
28
|
adantl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → 𝑥 ∈ ∪ ( 𝑅1 “ On ) ) |
30 |
|
uniwf |
⊢ ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ↔ ∪ 𝑥 ∈ ∪ ( 𝑅1 “ On ) ) |
31 |
29 30
|
sylib |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ∪ 𝑥 ∈ ∪ ( 𝑅1 “ On ) ) |
32 |
6
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → 𝐴 ∈ dom 𝑅1 ) |
33 |
|
rankr1ag |
⊢ ( ( ∪ 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐴 ∈ dom 𝑅1 ) → ( ∪ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ↔ ( rank ‘ ∪ 𝑥 ) ∈ 𝐴 ) ) |
34 |
31 32 33
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( ∪ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ↔ ( rank ‘ ∪ 𝑥 ) ∈ 𝐴 ) ) |
35 |
27 34
|
mpbird |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ∪ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) |
36 |
|
r1pwcl |
⊢ ( Lim 𝐴 → ( 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ↔ 𝒫 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ) |
37 |
36
|
adantl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) → ( 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ↔ 𝒫 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ) |
38 |
37
|
biimpa |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → 𝒫 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) |
39 |
28
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → 𝑥 ∈ ∪ ( 𝑅1 “ On ) ) |
40 |
|
r1elwf |
⊢ ( 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) → 𝑦 ∈ ∪ ( 𝑅1 “ On ) ) |
41 |
40
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → 𝑦 ∈ ∪ ( 𝑅1 “ On ) ) |
42 |
|
rankprb |
⊢ ( ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ) → ( rank ‘ { 𝑥 , 𝑦 } ) = suc ( ( rank ‘ 𝑥 ) ∪ ( rank ‘ 𝑦 ) ) ) |
43 |
39 41 42
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( rank ‘ { 𝑥 , 𝑦 } ) = suc ( ( rank ‘ 𝑥 ) ∪ ( rank ‘ 𝑦 ) ) ) |
44 |
|
limord |
⊢ ( Lim 𝐴 → Ord 𝐴 ) |
45 |
44
|
ad3antlr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → Ord 𝐴 ) |
46 |
20
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( rank ‘ 𝑥 ) ∈ 𝐴 ) |
47 |
|
rankr1ai |
⊢ ( 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) → ( rank ‘ 𝑦 ) ∈ 𝐴 ) |
48 |
47
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( rank ‘ 𝑦 ) ∈ 𝐴 ) |
49 |
|
ordunel |
⊢ ( ( Ord 𝐴 ∧ ( rank ‘ 𝑥 ) ∈ 𝐴 ∧ ( rank ‘ 𝑦 ) ∈ 𝐴 ) → ( ( rank ‘ 𝑥 ) ∪ ( rank ‘ 𝑦 ) ) ∈ 𝐴 ) |
50 |
45 46 48 49
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( ( rank ‘ 𝑥 ) ∪ ( rank ‘ 𝑦 ) ) ∈ 𝐴 ) |
51 |
|
limsuc |
⊢ ( Lim 𝐴 → ( ( ( rank ‘ 𝑥 ) ∪ ( rank ‘ 𝑦 ) ) ∈ 𝐴 ↔ suc ( ( rank ‘ 𝑥 ) ∪ ( rank ‘ 𝑦 ) ) ∈ 𝐴 ) ) |
52 |
51
|
ad3antlr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( ( ( rank ‘ 𝑥 ) ∪ ( rank ‘ 𝑦 ) ) ∈ 𝐴 ↔ suc ( ( rank ‘ 𝑥 ) ∪ ( rank ‘ 𝑦 ) ) ∈ 𝐴 ) ) |
53 |
50 52
|
mpbid |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → suc ( ( rank ‘ 𝑥 ) ∪ ( rank ‘ 𝑦 ) ) ∈ 𝐴 ) |
54 |
43 53
|
eqeltrd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( rank ‘ { 𝑥 , 𝑦 } ) ∈ 𝐴 ) |
55 |
|
prwf |
⊢ ( ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ) → { 𝑥 , 𝑦 } ∈ ∪ ( 𝑅1 “ On ) ) |
56 |
39 41 55
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → { 𝑥 , 𝑦 } ∈ ∪ ( 𝑅1 “ On ) ) |
57 |
32
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → 𝐴 ∈ dom 𝑅1 ) |
58 |
|
rankr1ag |
⊢ ( ( { 𝑥 , 𝑦 } ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐴 ∈ dom 𝑅1 ) → ( { 𝑥 , 𝑦 } ∈ ( 𝑅1 ‘ 𝐴 ) ↔ ( rank ‘ { 𝑥 , 𝑦 } ) ∈ 𝐴 ) ) |
59 |
56 57 58
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( { 𝑥 , 𝑦 } ∈ ( 𝑅1 ‘ 𝐴 ) ↔ ( rank ‘ { 𝑥 , 𝑦 } ) ∈ 𝐴 ) ) |
60 |
54 59
|
mpbird |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) ) → { 𝑥 , 𝑦 } ∈ ( 𝑅1 ‘ 𝐴 ) ) |
61 |
60
|
ralrimiva |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ∀ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) { 𝑥 , 𝑦 } ∈ ( 𝑅1 ‘ 𝐴 ) ) |
62 |
35 38 61
|
3jca |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) ∧ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) → ( ∪ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ∧ 𝒫 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) { 𝑥 , 𝑦 } ∈ ( 𝑅1 ‘ 𝐴 ) ) ) |
63 |
62
|
ralrimiva |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) → ∀ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ( ∪ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ∧ 𝒫 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) { 𝑥 , 𝑦 } ∈ ( 𝑅1 ‘ 𝐴 ) ) ) |
64 |
|
fvex |
⊢ ( 𝑅1 ‘ 𝐴 ) ∈ V |
65 |
|
iswun |
⊢ ( ( 𝑅1 ‘ 𝐴 ) ∈ V → ( ( 𝑅1 ‘ 𝐴 ) ∈ WUni ↔ ( Tr ( 𝑅1 ‘ 𝐴 ) ∧ ( 𝑅1 ‘ 𝐴 ) ≠ ∅ ∧ ∀ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ( ∪ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ∧ 𝒫 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) { 𝑥 , 𝑦 } ∈ ( 𝑅1 ‘ 𝐴 ) ) ) ) ) |
66 |
64 65
|
ax-mp |
⊢ ( ( 𝑅1 ‘ 𝐴 ) ∈ WUni ↔ ( Tr ( 𝑅1 ‘ 𝐴 ) ∧ ( 𝑅1 ‘ 𝐴 ) ≠ ∅ ∧ ∀ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ( ∪ 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ∧ 𝒫 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ ( 𝑅1 ‘ 𝐴 ) { 𝑥 , 𝑦 } ∈ ( 𝑅1 ‘ 𝐴 ) ) ) ) |
67 |
2 12 63 66
|
syl3anbrc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ Lim 𝐴 ) → ( 𝑅1 ‘ 𝐴 ) ∈ WUni ) |