Step |
Hyp |
Ref |
Expression |
1 |
|
rossros.q |
⊢ 𝑄 = { 𝑠 ∈ 𝒫 𝒫 𝑂 ∣ ( ∅ ∈ 𝑠 ∧ ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑠 ( ( 𝑥 ∪ 𝑦 ) ∈ 𝑠 ∧ ( 𝑥 ∖ 𝑦 ) ∈ 𝑠 ) ) } |
2 |
|
rossros.n |
⊢ 𝑁 = { 𝑠 ∈ 𝒫 𝒫 𝑂 ∣ ( ∅ ∈ 𝑠 ∧ ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑠 ( ( 𝑥 ∩ 𝑦 ) ∈ 𝑠 ∧ ∃ 𝑧 ∈ 𝒫 𝑠 ( 𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ ( 𝑥 ∖ 𝑦 ) = ∪ 𝑧 ) ) ) } |
3 |
1
|
rossspw |
⊢ ( 𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂 ) |
4 |
|
elpwg |
⊢ ( 𝑆 ∈ 𝑄 → ( 𝑆 ∈ 𝒫 𝒫 𝑂 ↔ 𝑆 ⊆ 𝒫 𝑂 ) ) |
5 |
3 4
|
mpbird |
⊢ ( 𝑆 ∈ 𝑄 → 𝑆 ∈ 𝒫 𝒫 𝑂 ) |
6 |
1
|
0elros |
⊢ ( 𝑆 ∈ 𝑄 → ∅ ∈ 𝑆 ) |
7 |
|
uneq1 |
⊢ ( 𝑢 = 𝑥 → ( 𝑢 ∪ 𝑣 ) = ( 𝑥 ∪ 𝑣 ) ) |
8 |
7
|
eleq1d |
⊢ ( 𝑢 = 𝑥 → ( ( 𝑢 ∪ 𝑣 ) ∈ 𝑠 ↔ ( 𝑥 ∪ 𝑣 ) ∈ 𝑠 ) ) |
9 |
|
difeq1 |
⊢ ( 𝑢 = 𝑥 → ( 𝑢 ∖ 𝑣 ) = ( 𝑥 ∖ 𝑣 ) ) |
10 |
9
|
eleq1d |
⊢ ( 𝑢 = 𝑥 → ( ( 𝑢 ∖ 𝑣 ) ∈ 𝑠 ↔ ( 𝑥 ∖ 𝑣 ) ∈ 𝑠 ) ) |
11 |
8 10
|
anbi12d |
⊢ ( 𝑢 = 𝑥 → ( ( ( 𝑢 ∪ 𝑣 ) ∈ 𝑠 ∧ ( 𝑢 ∖ 𝑣 ) ∈ 𝑠 ) ↔ ( ( 𝑥 ∪ 𝑣 ) ∈ 𝑠 ∧ ( 𝑥 ∖ 𝑣 ) ∈ 𝑠 ) ) ) |
12 |
|
uneq2 |
⊢ ( 𝑣 = 𝑦 → ( 𝑥 ∪ 𝑣 ) = ( 𝑥 ∪ 𝑦 ) ) |
13 |
12
|
eleq1d |
⊢ ( 𝑣 = 𝑦 → ( ( 𝑥 ∪ 𝑣 ) ∈ 𝑠 ↔ ( 𝑥 ∪ 𝑦 ) ∈ 𝑠 ) ) |
14 |
|
difeq2 |
⊢ ( 𝑣 = 𝑦 → ( 𝑥 ∖ 𝑣 ) = ( 𝑥 ∖ 𝑦 ) ) |
15 |
14
|
eleq1d |
⊢ ( 𝑣 = 𝑦 → ( ( 𝑥 ∖ 𝑣 ) ∈ 𝑠 ↔ ( 𝑥 ∖ 𝑦 ) ∈ 𝑠 ) ) |
16 |
13 15
|
anbi12d |
⊢ ( 𝑣 = 𝑦 → ( ( ( 𝑥 ∪ 𝑣 ) ∈ 𝑠 ∧ ( 𝑥 ∖ 𝑣 ) ∈ 𝑠 ) ↔ ( ( 𝑥 ∪ 𝑦 ) ∈ 𝑠 ∧ ( 𝑥 ∖ 𝑦 ) ∈ 𝑠 ) ) ) |
17 |
11 16
|
cbvral2vw |
⊢ ( ∀ 𝑢 ∈ 𝑠 ∀ 𝑣 ∈ 𝑠 ( ( 𝑢 ∪ 𝑣 ) ∈ 𝑠 ∧ ( 𝑢 ∖ 𝑣 ) ∈ 𝑠 ) ↔ ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑠 ( ( 𝑥 ∪ 𝑦 ) ∈ 𝑠 ∧ ( 𝑥 ∖ 𝑦 ) ∈ 𝑠 ) ) |
18 |
17
|
anbi2i |
⊢ ( ( ∅ ∈ 𝑠 ∧ ∀ 𝑢 ∈ 𝑠 ∀ 𝑣 ∈ 𝑠 ( ( 𝑢 ∪ 𝑣 ) ∈ 𝑠 ∧ ( 𝑢 ∖ 𝑣 ) ∈ 𝑠 ) ) ↔ ( ∅ ∈ 𝑠 ∧ ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑠 ( ( 𝑥 ∪ 𝑦 ) ∈ 𝑠 ∧ ( 𝑥 ∖ 𝑦 ) ∈ 𝑠 ) ) ) |
19 |
18
|
rabbii |
⊢ { 𝑠 ∈ 𝒫 𝒫 𝑂 ∣ ( ∅ ∈ 𝑠 ∧ ∀ 𝑢 ∈ 𝑠 ∀ 𝑣 ∈ 𝑠 ( ( 𝑢 ∪ 𝑣 ) ∈ 𝑠 ∧ ( 𝑢 ∖ 𝑣 ) ∈ 𝑠 ) ) } = { 𝑠 ∈ 𝒫 𝒫 𝑂 ∣ ( ∅ ∈ 𝑠 ∧ ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑠 ( ( 𝑥 ∪ 𝑦 ) ∈ 𝑠 ∧ ( 𝑥 ∖ 𝑦 ) ∈ 𝑠 ) ) } |
20 |
1 19
|
eqtr4i |
⊢ 𝑄 = { 𝑠 ∈ 𝒫 𝒫 𝑂 ∣ ( ∅ ∈ 𝑠 ∧ ∀ 𝑢 ∈ 𝑠 ∀ 𝑣 ∈ 𝑠 ( ( 𝑢 ∪ 𝑣 ) ∈ 𝑠 ∧ ( 𝑢 ∖ 𝑣 ) ∈ 𝑠 ) ) } |
21 |
20
|
inelros |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝑆 ) |
22 |
21
|
3expb |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝑆 ) |
23 |
20
|
difelros |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( 𝑥 ∖ 𝑦 ) ∈ 𝑆 ) |
24 |
23
|
3expb |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 ∖ 𝑦 ) ∈ 𝑆 ) |
25 |
24
|
snssd |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → { ( 𝑥 ∖ 𝑦 ) } ⊆ 𝑆 ) |
26 |
|
snex |
⊢ { ( 𝑥 ∖ 𝑦 ) } ∈ V |
27 |
26
|
elpw |
⊢ ( { ( 𝑥 ∖ 𝑦 ) } ∈ 𝒫 𝑆 ↔ { ( 𝑥 ∖ 𝑦 ) } ⊆ 𝑆 ) |
28 |
25 27
|
sylibr |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → { ( 𝑥 ∖ 𝑦 ) } ∈ 𝒫 𝑆 ) |
29 |
|
snfi |
⊢ { ( 𝑥 ∖ 𝑦 ) } ∈ Fin |
30 |
29
|
a1i |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → { ( 𝑥 ∖ 𝑦 ) } ∈ Fin ) |
31 |
|
disjxsn |
⊢ Disj 𝑡 ∈ { ( 𝑥 ∖ 𝑦 ) } 𝑡 |
32 |
31
|
a1i |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → Disj 𝑡 ∈ { ( 𝑥 ∖ 𝑦 ) } 𝑡 ) |
33 |
|
unisng |
⊢ ( ( 𝑥 ∖ 𝑦 ) ∈ 𝑆 → ∪ { ( 𝑥 ∖ 𝑦 ) } = ( 𝑥 ∖ 𝑦 ) ) |
34 |
24 33
|
syl |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ∪ { ( 𝑥 ∖ 𝑦 ) } = ( 𝑥 ∖ 𝑦 ) ) |
35 |
34
|
eqcomd |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 ∖ 𝑦 ) = ∪ { ( 𝑥 ∖ 𝑦 ) } ) |
36 |
|
eleq1 |
⊢ ( 𝑧 = { ( 𝑥 ∖ 𝑦 ) } → ( 𝑧 ∈ Fin ↔ { ( 𝑥 ∖ 𝑦 ) } ∈ Fin ) ) |
37 |
|
disjeq1 |
⊢ ( 𝑧 = { ( 𝑥 ∖ 𝑦 ) } → ( Disj 𝑡 ∈ 𝑧 𝑡 ↔ Disj 𝑡 ∈ { ( 𝑥 ∖ 𝑦 ) } 𝑡 ) ) |
38 |
|
unieq |
⊢ ( 𝑧 = { ( 𝑥 ∖ 𝑦 ) } → ∪ 𝑧 = ∪ { ( 𝑥 ∖ 𝑦 ) } ) |
39 |
38
|
eqeq2d |
⊢ ( 𝑧 = { ( 𝑥 ∖ 𝑦 ) } → ( ( 𝑥 ∖ 𝑦 ) = ∪ 𝑧 ↔ ( 𝑥 ∖ 𝑦 ) = ∪ { ( 𝑥 ∖ 𝑦 ) } ) ) |
40 |
36 37 39
|
3anbi123d |
⊢ ( 𝑧 = { ( 𝑥 ∖ 𝑦 ) } → ( ( 𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ ( 𝑥 ∖ 𝑦 ) = ∪ 𝑧 ) ↔ ( { ( 𝑥 ∖ 𝑦 ) } ∈ Fin ∧ Disj 𝑡 ∈ { ( 𝑥 ∖ 𝑦 ) } 𝑡 ∧ ( 𝑥 ∖ 𝑦 ) = ∪ { ( 𝑥 ∖ 𝑦 ) } ) ) ) |
41 |
40
|
rspcev |
⊢ ( ( { ( 𝑥 ∖ 𝑦 ) } ∈ 𝒫 𝑆 ∧ ( { ( 𝑥 ∖ 𝑦 ) } ∈ Fin ∧ Disj 𝑡 ∈ { ( 𝑥 ∖ 𝑦 ) } 𝑡 ∧ ( 𝑥 ∖ 𝑦 ) = ∪ { ( 𝑥 ∖ 𝑦 ) } ) ) → ∃ 𝑧 ∈ 𝒫 𝑆 ( 𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ ( 𝑥 ∖ 𝑦 ) = ∪ 𝑧 ) ) |
42 |
28 30 32 35 41
|
syl13anc |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ∃ 𝑧 ∈ 𝒫 𝑆 ( 𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ ( 𝑥 ∖ 𝑦 ) = ∪ 𝑧 ) ) |
43 |
22 42
|
jca |
⊢ ( ( 𝑆 ∈ 𝑄 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( ( 𝑥 ∩ 𝑦 ) ∈ 𝑆 ∧ ∃ 𝑧 ∈ 𝒫 𝑆 ( 𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ ( 𝑥 ∖ 𝑦 ) = ∪ 𝑧 ) ) ) |
44 |
43
|
ralrimivva |
⊢ ( 𝑆 ∈ 𝑄 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( ( 𝑥 ∩ 𝑦 ) ∈ 𝑆 ∧ ∃ 𝑧 ∈ 𝒫 𝑆 ( 𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ ( 𝑥 ∖ 𝑦 ) = ∪ 𝑧 ) ) ) |
45 |
5 6 44
|
3jca |
⊢ ( 𝑆 ∈ 𝑄 → ( 𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( ( 𝑥 ∩ 𝑦 ) ∈ 𝑆 ∧ ∃ 𝑧 ∈ 𝒫 𝑆 ( 𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ ( 𝑥 ∖ 𝑦 ) = ∪ 𝑧 ) ) ) ) |
46 |
2
|
issros |
⊢ ( 𝑆 ∈ 𝑁 ↔ ( 𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( ( 𝑥 ∩ 𝑦 ) ∈ 𝑆 ∧ ∃ 𝑧 ∈ 𝒫 𝑆 ( 𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ ( 𝑥 ∖ 𝑦 ) = ∪ 𝑧 ) ) ) ) |
47 |
45 46
|
sylibr |
⊢ ( 𝑆 ∈ 𝑄 → 𝑆 ∈ 𝑁 ) |