Step |
Hyp |
Ref |
Expression |
1 |
|
rankxplim.1 |
⊢ 𝐴 ∈ V |
2 |
|
rankxplim.2 |
⊢ 𝐵 ∈ V |
3 |
|
0ellim |
⊢ ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) → ∅ ∈ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
4 |
|
n0i |
⊢ ( ∅ ∈ ( rank ‘ ( 𝐴 × 𝐵 ) ) → ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
5 |
3 4
|
syl |
⊢ ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) → ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
6 |
|
df-ne |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ ↔ ¬ ( 𝐴 × 𝐵 ) = ∅ ) |
7 |
1 2
|
xpex |
⊢ ( 𝐴 × 𝐵 ) ∈ V |
8 |
7
|
rankeq0 |
⊢ ( ( 𝐴 × 𝐵 ) = ∅ ↔ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
9 |
8
|
notbii |
⊢ ( ¬ ( 𝐴 × 𝐵 ) = ∅ ↔ ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ) |
10 |
6 9
|
bitr2i |
⊢ ( ¬ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ∅ ↔ ( 𝐴 × 𝐵 ) ≠ ∅ ) |
11 |
5 10
|
sylib |
⊢ ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) → ( 𝐴 × 𝐵 ) ≠ ∅ ) |
12 |
|
limuni2 |
⊢ ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) → Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
13 |
|
limuni2 |
⊢ ( Lim ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) → Lim ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
14 |
12 13
|
syl |
⊢ ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) → Lim ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ) |
15 |
|
rankuni |
⊢ ( rank ‘ ∪ ∪ ( 𝐴 × 𝐵 ) ) = ∪ ( rank ‘ ∪ ( 𝐴 × 𝐵 ) ) |
16 |
|
rankuni |
⊢ ( rank ‘ ∪ ( 𝐴 × 𝐵 ) ) = ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) |
17 |
16
|
unieqi |
⊢ ∪ ( rank ‘ ∪ ( 𝐴 × 𝐵 ) ) = ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) |
18 |
15 17
|
eqtr2i |
⊢ ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ∪ ∪ ( 𝐴 × 𝐵 ) ) |
19 |
|
unixp |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ → ∪ ∪ ( 𝐴 × 𝐵 ) = ( 𝐴 ∪ 𝐵 ) ) |
20 |
19
|
fveq2d |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ → ( rank ‘ ∪ ∪ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
21 |
18 20
|
eqtrid |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ → ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
22 |
|
limeq |
⊢ ( ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) = ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) → ( Lim ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
23 |
21 22
|
syl |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ → ( Lim ∪ ∪ ( rank ‘ ( 𝐴 × 𝐵 ) ) ↔ Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
24 |
14 23
|
syl5ib |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ → ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) → Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) |
25 |
11 24
|
mpcom |
⊢ ( Lim ( rank ‘ ( 𝐴 × 𝐵 ) ) → Lim ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) |