Step |
Hyp |
Ref |
Expression |
1 |
|
rankxplim.1 |
⊢ 𝐴 ∈ V |
2 |
|
rankxplim.2 |
⊢ 𝐵 ∈ V |
3 |
|
limuni2 |
⊢ ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) → Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
4 |
|
0ellim |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ∅ ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
5 |
|
n0i |
⊢ ( ∅ ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ¬ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
6 |
|
unieq |
⊢ ( ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∪ ∅ ) |
7 |
|
uni0 |
⊢ ∪ ∅ = ∅ |
8 |
6 7
|
eqtrdi |
⊢ ( ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
9 |
8
|
con3i |
⊢ ( ¬ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
10 |
4 5 9
|
3syl |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
11 |
|
rankon |
⊢ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ On |
12 |
11
|
onsuci |
⊢ suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ On |
13 |
12
|
onsuci |
⊢ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ On |
14 |
13
|
elexi |
⊢ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ V |
15 |
14
|
sucid |
⊢ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) |
16 |
13
|
onsuci |
⊢ suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ On |
17 |
|
ontri1 |
⊢ ( ( suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ On ∧ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ On ) → ( suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ ¬ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
18 |
16 13 17
|
mp2an |
⊢ ( suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ ¬ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
19 |
18
|
con2bii |
⊢ ( suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ ¬ suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
20 |
15 19
|
mpbi |
⊢ ¬ suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) |
21 |
1 2
|
rankxpu |
⊢ ( rank ‘ ( 𝐴 × 𝐵 ) ) ⊆ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) |
22 |
|
sstr |
⊢ ( ( suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) ⊆ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) → suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
23 |
21 22
|
mpan2 |
⊢ ( suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 × 𝐵 ) ) → suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
24 |
20 23
|
mto |
⊢ ¬ suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 × 𝐵 ) ) |
25 |
|
reeanv |
⊢ ( ∃ 𝑥 ∈ On ∃ 𝑦 ∈ On ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ↔ ( ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) |
26 |
|
simprl |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ) |
27 |
|
simpr |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ) |
28 |
|
df-ne |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ ↔ ¬ ( 𝐴 × 𝐵 ) = ∅ ) |
29 |
1 2
|
xpex |
⊢ ( 𝐴 × 𝐵 ) ∈ V |
30 |
29
|
rankeq0 |
⊢ ( ( 𝐴 × 𝐵 ) = ∅ ↔ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
31 |
30
|
notbii |
⊢ ( ¬ ( 𝐴 × 𝐵 ) = ∅ ↔ ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
32 |
28 31
|
bitr2i |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ↔ ( 𝐴 × 𝐵 ) ≠ ∅ ) |
33 |
10 32
|
sylib |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( 𝐴 × 𝐵 ) ≠ ∅ ) |
34 |
|
unixp |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ → ∪ ∪ ( 𝐴 × 𝐵 ) = ( 𝐴 ∪ 𝐵 ) ) |
35 |
33 34
|
syl |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ∪ ∪ ( 𝐴 × 𝐵 ) = ( 𝐴 ∪ 𝐵 ) ) |
36 |
35
|
fveq2d |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( rank ‘ ∪ ∪ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
37 |
|
rankuni |
⊢ ( rank ‘ ∪ ∪ ( 𝐴 × 𝐵 ) ) = ∪ ( rank ‘ ∪ ( 𝐴 × 𝐵 ) ) |
38 |
|
rankuni |
⊢ ( rank ‘ ∪ ( 𝐴 × 𝐵 ) ) = ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) |
39 |
38
|
unieqi |
⊢ ∪ ( rank ‘ ∪ ( 𝐴 × 𝐵 ) ) = ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) |
40 |
37 39
|
eqtri |
⊢ ( rank ‘ ∪ ∪ ( 𝐴 × 𝐵 ) ) = ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) |
41 |
36 40
|
eqtr3di |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
42 |
|
eqimss |
⊢ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
43 |
41 42
|
syl |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
44 |
43
|
adantr |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
45 |
27 44
|
eqsstrrd |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ) → suc 𝑥 ⊆ ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
46 |
45
|
adantrr |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → suc 𝑥 ⊆ ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
47 |
|
limuni |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
48 |
47
|
adantr |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
49 |
46 48
|
sseqtrrd |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → suc 𝑥 ⊆ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
50 |
|
vex |
⊢ 𝑥 ∈ V |
51 |
|
rankon |
⊢ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∈ On |
52 |
51
|
onordi |
⊢ Ord ( rank ‘ ( 𝐴 × 𝐵 ) ) |
53 |
|
orduni |
⊢ ( Ord ( rank ‘ ( 𝐴 × 𝐵 ) ) → Ord ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
54 |
52 53
|
ax-mp |
⊢ Ord ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) |
55 |
|
ordelsuc |
⊢ ( ( 𝑥 ∈ V ∧ Ord ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) → ( 𝑥 ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ suc 𝑥 ⊆ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
56 |
50 54 55
|
mp2an |
⊢ ( 𝑥 ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ suc 𝑥 ⊆ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
57 |
49 56
|
sylibr |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → 𝑥 ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
58 |
|
limsuc |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( 𝑥 ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ suc 𝑥 ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
59 |
58
|
adantr |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → ( 𝑥 ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ suc 𝑥 ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
60 |
57 59
|
mpbid |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → suc 𝑥 ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
61 |
26 60
|
eqeltrd |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
62 |
|
limsuc |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
63 |
62
|
adantr |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
64 |
61 63
|
mpbid |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
65 |
|
ordsucelsuc |
⊢ ( Ord ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ suc ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
66 |
54 65
|
ax-mp |
⊢ ( suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ suc ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
67 |
64 66
|
sylib |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ suc ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
68 |
|
onsucuni2 |
⊢ ( ( ( rank ‘ ( 𝐴 × 𝐵 ) ) ∈ On ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) → suc ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
69 |
51 68
|
mpan |
⊢ ( ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 → suc ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
70 |
69
|
ad2antll |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → suc ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
71 |
67 70
|
eleqtrd |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
72 |
13 51
|
onsucssi |
⊢ ( suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
73 |
71 72
|
sylib |
⊢ ( ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ∧ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
74 |
73
|
ex |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) → suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
75 |
74
|
a1d |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) → ( ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) → suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) ) |
76 |
75
|
rexlimdvv |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( ∃ 𝑥 ∈ On ∃ 𝑦 ∈ On ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) → suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
77 |
25 76
|
syl5bir |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( ( ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) → suc suc suc ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
78 |
24 77
|
mtoi |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ¬ ( ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) |
79 |
|
ianor |
⊢ ( ¬ ( ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ↔ ( ¬ ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∨ ¬ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) |
80 |
|
un00 |
⊢ ( ( 𝐴 = ∅ ∧ 𝐵 = ∅ ) ↔ ( 𝐴 ∪ 𝐵 ) = ∅ ) |
81 |
|
animorl |
⊢ ( ( 𝐴 = ∅ ∧ 𝐵 = ∅ ) → ( 𝐴 = ∅ ∨ 𝐵 = ∅ ) ) |
82 |
80 81
|
sylbir |
⊢ ( ( 𝐴 ∪ 𝐵 ) = ∅ → ( 𝐴 = ∅ ∨ 𝐵 = ∅ ) ) |
83 |
|
xpeq0 |
⊢ ( ( 𝐴 × 𝐵 ) = ∅ ↔ ( 𝐴 = ∅ ∨ 𝐵 = ∅ ) ) |
84 |
82 83
|
sylibr |
⊢ ( ( 𝐴 ∪ 𝐵 ) = ∅ → ( 𝐴 × 𝐵 ) = ∅ ) |
85 |
84
|
con3i |
⊢ ( ¬ ( 𝐴 × 𝐵 ) = ∅ → ¬ ( 𝐴 ∪ 𝐵 ) = ∅ ) |
86 |
31 85
|
sylbir |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ¬ ( 𝐴 ∪ 𝐵 ) = ∅ ) |
87 |
1 2
|
unex |
⊢ ( 𝐴 ∪ 𝐵 ) ∈ V |
88 |
87
|
rankeq0 |
⊢ ( ( 𝐴 ∪ 𝐵 ) = ∅ ↔ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ∅ ) |
89 |
88
|
notbii |
⊢ ( ¬ ( 𝐴 ∪ 𝐵 ) = ∅ ↔ ¬ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ∅ ) |
90 |
86 89
|
sylib |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ¬ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ∅ ) |
91 |
11
|
onordi |
⊢ Ord ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) |
92 |
|
ordzsl |
⊢ ( Ord ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ∅ ∨ ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∨ Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
93 |
91 92
|
mpbi |
⊢ ( ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ∅ ∨ ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∨ Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
94 |
93
|
3ori |
⊢ ( ( ¬ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ∅ ∧ ¬ ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ) → Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
95 |
90 94
|
sylan |
⊢ ( ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ∧ ¬ ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ) → Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
96 |
95
|
ex |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ( ¬ ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 → Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
97 |
|
ordzsl |
⊢ ( Ord ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ ( ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ∨ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ∨ Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
98 |
52 97
|
mpbi |
⊢ ( ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ∨ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ∨ Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
99 |
98
|
3ori |
⊢ ( ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ∧ ¬ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) → Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
100 |
99
|
ex |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ( ¬ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 → Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
101 |
96 100
|
orim12d |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ( ( ¬ ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∨ ¬ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) → ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∨ Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) ) |
102 |
79 101
|
syl5bi |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ( ¬ ( ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) → ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∨ Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) ) |
103 |
102
|
imp |
⊢ ( ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ∧ ¬ ( ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∨ Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
104 |
|
simpl |
⊢ ( ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∧ ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) → Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
105 |
30
|
necon3abii |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ ↔ ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
106 |
1 2
|
rankxplim |
⊢ ( ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∧ ( 𝐴 × 𝐵 ) ≠ ∅ ) → ( rank ‘ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
107 |
105 106
|
sylan2br |
⊢ ( ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∧ ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) → ( rank ‘ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
108 |
|
limeq |
⊢ ( ( rank ‘ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) → ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
109 |
107 108
|
syl |
⊢ ( ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∧ ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) → ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
110 |
104 109
|
mpbird |
⊢ ( ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∧ ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) → Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
111 |
110
|
expcom |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) → Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
112 |
|
idd |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) → Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
113 |
111 112
|
jaod |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ → ( ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∨ Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) → Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
114 |
113
|
adantr |
⊢ ( ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ∧ ¬ ( ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → ( ( Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∨ Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) → Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) ) |
115 |
103 114
|
mpd |
⊢ ( ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ∧ ¬ ( ∃ 𝑥 ∈ On ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = suc 𝑥 ∧ ∃ 𝑦 ∈ On ( rank ‘ ( 𝐴 × 𝐵 ) ) = suc 𝑦 ) ) → Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
116 |
10 78 115
|
syl2anc |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
117 |
3 116
|
impbii |
⊢ ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |