Step |
Hyp |
Ref |
Expression |
1 |
|
chpssat.1 |
⊢ 𝐴 ∈ Cℋ |
2 |
|
chpssat.2 |
⊢ 𝐵 ∈ Cℋ |
3 |
|
nssinpss |
⊢ ( ¬ 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) ⊊ 𝐴 ) |
4 |
1 2
|
chincli |
⊢ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ |
5 |
4 1
|
chrelati |
⊢ ( ( 𝐴 ∩ 𝐵 ) ⊊ 𝐴 → ∃ 𝑥 ∈ HAtoms ( ( 𝐴 ∩ 𝐵 ) ⊊ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ⊆ 𝐴 ) ) |
6 |
|
atelch |
⊢ ( 𝑥 ∈ HAtoms → 𝑥 ∈ Cℋ ) |
7 |
|
chlub |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∈ Cℋ ∧ 𝑥 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴 ) ↔ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ⊆ 𝐴 ) ) |
8 |
4 1 7
|
mp3an13 |
⊢ ( 𝑥 ∈ Cℋ → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴 ) ↔ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ⊆ 𝐴 ) ) |
9 |
|
simpr |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴 ) → 𝑥 ⊆ 𝐴 ) |
10 |
8 9
|
syl6bir |
⊢ ( 𝑥 ∈ Cℋ → ( ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ⊆ 𝐴 → 𝑥 ⊆ 𝐴 ) ) |
11 |
10
|
adantld |
⊢ ( 𝑥 ∈ Cℋ → ( ( ( 𝐴 ∩ 𝐵 ) ⊊ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ⊆ 𝐴 ) → 𝑥 ⊆ 𝐴 ) ) |
12 |
|
ssin |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ↔ 𝑥 ⊆ ( 𝐴 ∩ 𝐵 ) ) |
13 |
12
|
notbii |
⊢ ( ¬ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ↔ ¬ 𝑥 ⊆ ( 𝐴 ∩ 𝐵 ) ) |
14 |
|
chnle |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → ( ¬ 𝑥 ⊆ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝐴 ∩ 𝐵 ) ⊊ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ) ) |
15 |
4 14
|
mpan |
⊢ ( 𝑥 ∈ Cℋ → ( ¬ 𝑥 ⊆ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝐴 ∩ 𝐵 ) ⊊ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ) ) |
16 |
13 15
|
syl5bb |
⊢ ( 𝑥 ∈ Cℋ → ( ¬ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ↔ ( 𝐴 ∩ 𝐵 ) ⊊ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ) ) |
17 |
16 8
|
anbi12d |
⊢ ( 𝑥 ∈ Cℋ → ( ( ¬ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴 ) ) ↔ ( ( 𝐴 ∩ 𝐵 ) ⊊ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ⊆ 𝐴 ) ) ) |
18 |
|
pm3.21 |
⊢ ( 𝑥 ⊆ 𝐵 → ( 𝑥 ⊆ 𝐴 → ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ) ) |
19 |
|
orcom |
⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ∨ ¬ 𝑥 ⊆ 𝐴 ) ↔ ( ¬ 𝑥 ⊆ 𝐴 ∨ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ) ) |
20 |
|
pm4.55 |
⊢ ( ¬ ( ¬ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ∧ 𝑥 ⊆ 𝐴 ) ↔ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ∨ ¬ 𝑥 ⊆ 𝐴 ) ) |
21 |
|
imor |
⊢ ( ( 𝑥 ⊆ 𝐴 → ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ) ↔ ( ¬ 𝑥 ⊆ 𝐴 ∨ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ) ) |
22 |
19 20 21
|
3bitr4ri |
⊢ ( ( 𝑥 ⊆ 𝐴 → ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ) ↔ ¬ ( ¬ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ∧ 𝑥 ⊆ 𝐴 ) ) |
23 |
18 22
|
sylib |
⊢ ( 𝑥 ⊆ 𝐵 → ¬ ( ¬ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ∧ 𝑥 ⊆ 𝐴 ) ) |
24 |
23
|
con2i |
⊢ ( ( ¬ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ∧ 𝑥 ⊆ 𝐴 ) → ¬ 𝑥 ⊆ 𝐵 ) |
25 |
24
|
adantrl |
⊢ ( ( ¬ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴 ) ) → ¬ 𝑥 ⊆ 𝐵 ) |
26 |
17 25
|
syl6bir |
⊢ ( 𝑥 ∈ Cℋ → ( ( ( 𝐴 ∩ 𝐵 ) ⊊ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ⊆ 𝐴 ) → ¬ 𝑥 ⊆ 𝐵 ) ) |
27 |
11 26
|
jcad |
⊢ ( 𝑥 ∈ Cℋ → ( ( ( 𝐴 ∩ 𝐵 ) ⊊ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ⊆ 𝐴 ) → ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) ) |
28 |
6 27
|
syl |
⊢ ( 𝑥 ∈ HAtoms → ( ( ( 𝐴 ∩ 𝐵 ) ⊊ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ⊆ 𝐴 ) → ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) ) |
29 |
28
|
reximia |
⊢ ( ∃ 𝑥 ∈ HAtoms ( ( 𝐴 ∩ 𝐵 ) ⊊ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝑥 ) ⊆ 𝐴 ) → ∃ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
30 |
5 29
|
syl |
⊢ ( ( 𝐴 ∩ 𝐵 ) ⊊ 𝐴 → ∃ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
31 |
3 30
|
sylbi |
⊢ ( ¬ 𝐴 ⊆ 𝐵 → ∃ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
32 |
|
sstr2 |
⊢ ( 𝑥 ⊆ 𝐴 → ( 𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵 ) ) |
33 |
32
|
com12 |
⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ) |
34 |
33
|
ralrimivw |
⊢ ( 𝐴 ⊆ 𝐵 → ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ) |
35 |
|
iman |
⊢ ( ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ↔ ¬ ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
36 |
35
|
ralbii |
⊢ ( ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ↔ ∀ 𝑥 ∈ HAtoms ¬ ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
37 |
|
ralnex |
⊢ ( ∀ 𝑥 ∈ HAtoms ¬ ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ↔ ¬ ∃ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
38 |
36 37
|
bitri |
⊢ ( ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ↔ ¬ ∃ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
39 |
34 38
|
sylib |
⊢ ( 𝐴 ⊆ 𝐵 → ¬ ∃ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
40 |
39
|
con2i |
⊢ ( ∃ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) → ¬ 𝐴 ⊆ 𝐵 ) |
41 |
31 40
|
impbii |
⊢ ( ¬ 𝐴 ⊆ 𝐵 ↔ ∃ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |