Step |
Hyp |
Ref |
Expression |
1 |
|
ist0cls.1 |
⊢ ( 𝜑 → 𝐵 = ∪ 𝐽 ) |
2 |
|
ist0cls.2 |
⊢ ( 𝜑 → 𝐷 = ( Clsd ‘ 𝐽 ) ) |
3 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
4 |
3
|
ist0 |
⊢ ( 𝐽 ∈ Kol2 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) ) |
5 |
4
|
simplbi |
⊢ ( 𝐽 ∈ Kol2 → 𝐽 ∈ Top ) |
6 |
5
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ Kol2 ) → 𝐽 ∈ Top ) |
7 |
4
|
baib |
⊢ ( 𝐽 ∈ Top → ( 𝐽 ∈ Kol2 ↔ ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) ) |
8 |
7
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ Top ) → ( 𝐽 ∈ Kol2 ↔ ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) ) |
9 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ Top ) → 𝐵 = ∪ 𝐽 ) |
10 |
9
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ Top ) → ∪ 𝐽 = 𝐵 ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) → ∪ 𝐽 = 𝐵 ) |
12 |
|
simp-4r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑜 ∈ 𝐽 ) → 𝐽 ∈ Top ) |
13 |
|
uniexg |
⊢ ( 𝐽 ∈ Top → ∪ 𝐽 ∈ V ) |
14 |
|
difexg |
⊢ ( ∪ 𝐽 ∈ V → ( ∪ 𝐽 ∖ 𝑜 ) ∈ V ) |
15 |
12 13 14
|
3syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑜 ∈ 𝐽 ) → ( ∪ 𝐽 ∖ 𝑜 ) ∈ V ) |
16 |
3
|
iscld |
⊢ ( 𝐽 ∈ Top → ( 𝑑 ∈ ( Clsd ‘ 𝐽 ) ↔ ( 𝑑 ⊆ ∪ 𝐽 ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ Top ) → ( 𝑑 ∈ ( Clsd ‘ 𝐽 ) ↔ ( 𝑑 ⊆ ∪ 𝐽 ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) ) ) |
18 |
2
|
eleq2d |
⊢ ( 𝜑 → ( 𝑑 ∈ 𝐷 ↔ 𝑑 ∈ ( Clsd ‘ 𝐽 ) ) ) |
19 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ Top ) → ( 𝑑 ∈ 𝐷 ↔ 𝑑 ∈ ( Clsd ‘ 𝐽 ) ) ) |
20 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑜 ∈ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) |
21 |
|
difssd |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑜 ∈ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( ∪ 𝐽 ∖ 𝑜 ) ⊆ ∪ 𝐽 ) |
22 |
20 21
|
eqsstrd |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑜 ∈ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → 𝑑 ⊆ ∪ 𝐽 ) |
23 |
22
|
r19.29an |
⊢ ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ ∃ 𝑜 ∈ 𝐽 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → 𝑑 ⊆ ∪ 𝐽 ) |
24 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ 𝑜 ∈ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) |
25 |
24
|
difeq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ 𝑜 ∈ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( ∪ 𝐽 ∖ 𝑑 ) = ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ 𝑜 ) ) ) |
26 |
3
|
eltopss |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → 𝑜 ⊆ ∪ 𝐽 ) |
27 |
26
|
ad5ant24 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ 𝑜 ∈ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → 𝑜 ⊆ ∪ 𝐽 ) |
28 |
|
dfss4 |
⊢ ( 𝑜 ⊆ ∪ 𝐽 ↔ ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ 𝑜 ) ) = 𝑜 ) |
29 |
27 28
|
sylib |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ 𝑜 ∈ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ 𝑜 ) ) = 𝑜 ) |
30 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ 𝑜 ∈ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → 𝑜 ∈ 𝐽 ) |
31 |
29 30
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ 𝑜 ∈ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ 𝑜 ) ) ∈ 𝐽 ) |
32 |
25 31
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ 𝑜 ∈ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) |
33 |
32
|
r19.29an |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ ∃ 𝑜 ∈ 𝐽 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) |
34 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) → ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) |
35 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) ∧ 𝑜 = ( ∪ 𝐽 ∖ 𝑑 ) ) → 𝑜 = ( ∪ 𝐽 ∖ 𝑑 ) ) |
36 |
35
|
difeq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) ∧ 𝑜 = ( ∪ 𝐽 ∖ 𝑑 ) ) → ( ∪ 𝐽 ∖ 𝑜 ) = ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ 𝑑 ) ) ) |
37 |
36
|
eqeq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) ∧ 𝑜 = ( ∪ 𝐽 ∖ 𝑑 ) ) → ( 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ↔ 𝑑 = ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ 𝑑 ) ) ) ) |
38 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) → 𝑑 ⊆ ∪ 𝐽 ) |
39 |
|
dfss4 |
⊢ ( 𝑑 ⊆ ∪ 𝐽 ↔ ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ 𝑑 ) ) = 𝑑 ) |
40 |
38 39
|
sylib |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) → ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ 𝑑 ) ) = 𝑑 ) |
41 |
40
|
eqcomd |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) → 𝑑 = ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ 𝑑 ) ) ) |
42 |
34 37 41
|
rspcedvd |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) → ∃ 𝑜 ∈ 𝐽 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) |
43 |
33 42
|
impbida |
⊢ ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑑 ⊆ ∪ 𝐽 ) → ( ∃ 𝑜 ∈ 𝐽 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ↔ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) ) |
44 |
23 43
|
biadanid |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ Top ) → ( ∃ 𝑜 ∈ 𝐽 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ↔ ( 𝑑 ⊆ ∪ 𝐽 ∧ ( ∪ 𝐽 ∖ 𝑑 ) ∈ 𝐽 ) ) ) |
45 |
17 19 44
|
3bitr4d |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ Top ) → ( 𝑑 ∈ 𝐷 ↔ ∃ 𝑜 ∈ 𝐽 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) ) |
46 |
45
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) → ( 𝑑 ∈ 𝐷 ↔ ∃ 𝑜 ∈ 𝐽 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) ) |
47 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) |
48 |
47
|
eleq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( 𝑥 ∈ 𝑑 ↔ 𝑥 ∈ ( ∪ 𝐽 ∖ 𝑜 ) ) ) |
49 |
|
eldif |
⊢ ( 𝑥 ∈ ( ∪ 𝐽 ∖ 𝑜 ) ↔ ( 𝑥 ∈ ∪ 𝐽 ∧ ¬ 𝑥 ∈ 𝑜 ) ) |
50 |
49
|
baib |
⊢ ( 𝑥 ∈ ∪ 𝐽 → ( 𝑥 ∈ ( ∪ 𝐽 ∖ 𝑜 ) ↔ ¬ 𝑥 ∈ 𝑜 ) ) |
51 |
50
|
ad3antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( 𝑥 ∈ ( ∪ 𝐽 ∖ 𝑜 ) ↔ ¬ 𝑥 ∈ 𝑜 ) ) |
52 |
48 51
|
bitrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( 𝑥 ∈ 𝑑 ↔ ¬ 𝑥 ∈ 𝑜 ) ) |
53 |
47
|
eleq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( 𝑦 ∈ 𝑑 ↔ 𝑦 ∈ ( ∪ 𝐽 ∖ 𝑜 ) ) ) |
54 |
|
eldif |
⊢ ( 𝑦 ∈ ( ∪ 𝐽 ∖ 𝑜 ) ↔ ( 𝑦 ∈ ∪ 𝐽 ∧ ¬ 𝑦 ∈ 𝑜 ) ) |
55 |
54
|
baib |
⊢ ( 𝑦 ∈ ∪ 𝐽 → ( 𝑦 ∈ ( ∪ 𝐽 ∖ 𝑜 ) ↔ ¬ 𝑦 ∈ 𝑜 ) ) |
56 |
55
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( 𝑦 ∈ ( ∪ 𝐽 ∖ 𝑜 ) ↔ ¬ 𝑦 ∈ 𝑜 ) ) |
57 |
53 56
|
bitrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( 𝑦 ∈ 𝑑 ↔ ¬ 𝑦 ∈ 𝑜 ) ) |
58 |
52 57
|
bibi12d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( ( 𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑 ) ↔ ( ¬ 𝑥 ∈ 𝑜 ↔ ¬ 𝑦 ∈ 𝑜 ) ) ) |
59 |
|
notbi |
⊢ ( ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ↔ ( ¬ 𝑥 ∈ 𝑜 ↔ ¬ 𝑦 ∈ 𝑜 ) ) |
60 |
58 59
|
bitr4di |
⊢ ( ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) ∧ 𝑑 = ( ∪ 𝐽 ∖ 𝑜 ) ) → ( ( 𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑 ) ↔ ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) ) |
61 |
15 46 60
|
ralxfr2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) → ( ∀ 𝑑 ∈ 𝐷 ( 𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑 ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) ) |
62 |
61
|
bicomd |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) → ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ↔ ∀ 𝑑 ∈ 𝐷 ( 𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑 ) ) ) |
63 |
62
|
imbi1d |
⊢ ( ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) ∧ 𝑦 ∈ ∪ 𝐽 ) → ( ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ↔ ( ∀ 𝑑 ∈ 𝐷 ( 𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑 ) → 𝑥 = 𝑦 ) ) ) |
64 |
11 63
|
raleqbidva |
⊢ ( ( ( 𝜑 ∧ 𝐽 ∈ Top ) ∧ 𝑥 ∈ ∪ 𝐽 ) → ( ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑑 ∈ 𝐷 ( 𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑 ) → 𝑥 = 𝑦 ) ) ) |
65 |
10 64
|
raleqbidva |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ Top ) → ( ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑑 ∈ 𝐷 ( 𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑 ) → 𝑥 = 𝑦 ) ) ) |
66 |
8 65
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ Top ) → ( 𝐽 ∈ Kol2 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑑 ∈ 𝐷 ( 𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑 ) → 𝑥 = 𝑦 ) ) ) |
67 |
6 66
|
biadanid |
⊢ ( 𝜑 → ( 𝐽 ∈ Kol2 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑑 ∈ 𝐷 ( 𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑 ) → 𝑥 = 𝑦 ) ) ) ) |