Step |
Hyp |
Ref |
Expression |
1 |
|
cfslb.1 |
⊢ 𝐴 ∈ V |
2 |
|
fvex |
⊢ ( card ‘ 𝐵 ) ∈ V |
3 |
|
ssid |
⊢ ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐵 ) |
4 |
1
|
ssex |
⊢ ( 𝐵 ⊆ 𝐴 → 𝐵 ∈ V ) |
5 |
4
|
ad2antrr |
⊢ ( ( ( 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) ∧ ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐵 ) ) → 𝐵 ∈ V ) |
6 |
|
velpw |
⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 ) |
7 |
|
sseq1 |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴 ) ) |
8 |
6 7
|
syl5bb |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴 ) ) |
9 |
|
unieq |
⊢ ( 𝑥 = 𝐵 → ∪ 𝑥 = ∪ 𝐵 ) |
10 |
9
|
eqeq1d |
⊢ ( 𝑥 = 𝐵 → ( ∪ 𝑥 = 𝐴 ↔ ∪ 𝐵 = 𝐴 ) ) |
11 |
8 10
|
anbi12d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴 ) ↔ ( 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) ) ) |
12 |
|
fveq2 |
⊢ ( 𝑥 = 𝐵 → ( card ‘ 𝑥 ) = ( card ‘ 𝐵 ) ) |
13 |
12
|
sseq1d |
⊢ ( 𝑥 = 𝐵 → ( ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ↔ ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐵 ) ) ) |
14 |
11 13
|
anbi12d |
⊢ ( 𝑥 = 𝐵 → ( ( ( 𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴 ) ∧ ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) ↔ ( ( 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) ∧ ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐵 ) ) ) ) |
15 |
14
|
spcegv |
⊢ ( 𝐵 ∈ V → ( ( ( 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) ∧ ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐵 ) ) → ∃ 𝑥 ( ( 𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴 ) ∧ ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) ) ) |
16 |
5 15
|
mpcom |
⊢ ( ( ( 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) ∧ ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐵 ) ) → ∃ 𝑥 ( ( 𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴 ) ∧ ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) ) |
17 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ↔ ∃ 𝑥 ( 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ∧ ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) ) |
18 |
|
rabid |
⊢ ( 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ↔ ( 𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴 ) ) |
19 |
18
|
anbi1i |
⊢ ( ( 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ∧ ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) ↔ ( ( 𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴 ) ∧ ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) ) |
20 |
19
|
exbii |
⊢ ( ∃ 𝑥 ( 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ∧ ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) ↔ ∃ 𝑥 ( ( 𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴 ) ∧ ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) ) |
21 |
17 20
|
bitri |
⊢ ( ∃ 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ↔ ∃ 𝑥 ( ( 𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴 ) ∧ ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) ) |
22 |
16 21
|
sylibr |
⊢ ( ( ( 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) ∧ ( card ‘ 𝐵 ) ⊆ ( card ‘ 𝐵 ) ) → ∃ 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) |
23 |
3 22
|
mpan2 |
⊢ ( ( 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) → ∃ 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) |
24 |
|
iinss |
⊢ ( ∃ 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) → ∩ 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) |
25 |
23 24
|
syl |
⊢ ( ( 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) → ∩ 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) |
26 |
1
|
cflim3 |
⊢ ( Lim 𝐴 → ( cf ‘ 𝐴 ) = ∩ 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ( card ‘ 𝑥 ) ) |
27 |
26
|
sseq1d |
⊢ ( Lim 𝐴 → ( ( cf ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ↔ ∩ 𝑥 ∈ { 𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴 } ( card ‘ 𝑥 ) ⊆ ( card ‘ 𝐵 ) ) ) |
28 |
25 27
|
syl5ibr |
⊢ ( Lim 𝐴 → ( ( 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) → ( cf ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ) ) |
29 |
28
|
3impib |
⊢ ( ( Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) → ( cf ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ) |
30 |
|
ssdomg |
⊢ ( ( card ‘ 𝐵 ) ∈ V → ( ( cf ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) → ( cf ‘ 𝐴 ) ≼ ( card ‘ 𝐵 ) ) ) |
31 |
2 29 30
|
mpsyl |
⊢ ( ( Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) → ( cf ‘ 𝐴 ) ≼ ( card ‘ 𝐵 ) ) |
32 |
|
limord |
⊢ ( Lim 𝐴 → Ord 𝐴 ) |
33 |
|
ordsson |
⊢ ( Ord 𝐴 → 𝐴 ⊆ On ) |
34 |
32 33
|
syl |
⊢ ( Lim 𝐴 → 𝐴 ⊆ On ) |
35 |
|
sstr2 |
⊢ ( 𝐵 ⊆ 𝐴 → ( 𝐴 ⊆ On → 𝐵 ⊆ On ) ) |
36 |
34 35
|
mpan9 |
⊢ ( ( Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ⊆ On ) |
37 |
|
onssnum |
⊢ ( ( 𝐵 ∈ V ∧ 𝐵 ⊆ On ) → 𝐵 ∈ dom card ) |
38 |
4 36 37
|
syl2an2 |
⊢ ( ( Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ∈ dom card ) |
39 |
|
cardid2 |
⊢ ( 𝐵 ∈ dom card → ( card ‘ 𝐵 ) ≈ 𝐵 ) |
40 |
38 39
|
syl |
⊢ ( ( Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( card ‘ 𝐵 ) ≈ 𝐵 ) |
41 |
40
|
3adant3 |
⊢ ( ( Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) → ( card ‘ 𝐵 ) ≈ 𝐵 ) |
42 |
|
domentr |
⊢ ( ( ( cf ‘ 𝐴 ) ≼ ( card ‘ 𝐵 ) ∧ ( card ‘ 𝐵 ) ≈ 𝐵 ) → ( cf ‘ 𝐴 ) ≼ 𝐵 ) |
43 |
31 41 42
|
syl2anc |
⊢ ( ( Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴 ) → ( cf ‘ 𝐴 ) ≼ 𝐵 ) |