Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) → 𝑉 ∈ V ) |
2 |
|
eleq1 |
⊢ ( 𝑐 = { 𝑥 , 𝑦 } → ( 𝑐 ∈ 𝑃 ↔ { 𝑥 , 𝑦 } ∈ 𝑃 ) ) |
3 |
|
prsssprel |
⊢ ( ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ∧ { 𝑥 , 𝑦 } ∈ 𝑃 ∧ ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) |
4 |
3
|
3exp |
⊢ ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → ( { 𝑥 , 𝑦 } ∈ 𝑃 → ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) ) |
5 |
4
|
com13 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) → ( { 𝑥 , 𝑦 } ∈ 𝑃 → ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) ) |
6 |
5
|
el2v |
⊢ ( { 𝑥 , 𝑦 } ∈ 𝑃 → ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) |
7 |
2 6
|
syl6bi |
⊢ ( 𝑐 = { 𝑥 , 𝑦 } → ( 𝑐 ∈ 𝑃 → ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) ) |
8 |
7
|
com12 |
⊢ ( 𝑐 ∈ 𝑃 → ( 𝑐 = { 𝑥 , 𝑦 } → ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) ) |
9 |
8
|
rexlimiv |
⊢ ( ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } → ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) |
10 |
9
|
com12 |
⊢ ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → ( ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) |
11 |
10
|
adantl |
⊢ ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) → ( ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) |
12 |
11
|
imp |
⊢ ( ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) |
13 |
12
|
simpld |
⊢ ( ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) → 𝑥 ∈ 𝑉 ) |
14 |
12
|
simprd |
⊢ ( ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) → 𝑦 ∈ 𝑉 ) |
15 |
1 1 13 14
|
opabex2 |
⊢ ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) → { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } ∈ V ) |
16 |
|
elopab |
⊢ ( 𝑝 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) ) |
17 |
9
|
adantl |
⊢ ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) → ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) |
18 |
17
|
adantld |
⊢ ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) → ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) |
19 |
18
|
imp |
⊢ ( ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) ∧ ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) |
20 |
|
eleq1 |
⊢ ( 𝑝 = 〈 𝑥 , 𝑦 〉 → ( 𝑝 ∈ ( 𝑉 × 𝑉 ) ↔ 〈 𝑥 , 𝑦 〉 ∈ ( 𝑉 × 𝑉 ) ) ) |
21 |
20
|
ad2antrr |
⊢ ( ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) ∧ ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) ) → ( 𝑝 ∈ ( 𝑉 × 𝑉 ) ↔ 〈 𝑥 , 𝑦 〉 ∈ ( 𝑉 × 𝑉 ) ) ) |
22 |
|
opelxp |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝑉 × 𝑉 ) ↔ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) |
23 |
21 22
|
bitrdi |
⊢ ( ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) ∧ ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) ) → ( 𝑝 ∈ ( 𝑉 × 𝑉 ) ↔ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ) |
24 |
19 23
|
mpbird |
⊢ ( ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) ∧ ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) ) → 𝑝 ∈ ( 𝑉 × 𝑉 ) ) |
25 |
24
|
ex |
⊢ ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) → ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) → 𝑝 ∈ ( 𝑉 × 𝑉 ) ) ) |
26 |
25
|
exlimivv |
⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) → ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) → 𝑝 ∈ ( 𝑉 × 𝑉 ) ) ) |
27 |
16 26
|
sylbi |
⊢ ( 𝑝 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } → ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) → 𝑝 ∈ ( 𝑉 × 𝑉 ) ) ) |
28 |
27
|
com12 |
⊢ ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) → ( 𝑝 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } → 𝑝 ∈ ( 𝑉 × 𝑉 ) ) ) |
29 |
28
|
ssrdv |
⊢ ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) → { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } ⊆ ( 𝑉 × 𝑉 ) ) |
30 |
15 29
|
elpwd |
⊢ ( ( 𝑉 ∈ V ∧ 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ) → { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } ∈ 𝒫 ( 𝑉 × 𝑉 ) ) |
31 |
30
|
ex |
⊢ ( 𝑉 ∈ V → ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } ∈ 𝒫 ( 𝑉 × 𝑉 ) ) ) |
32 |
|
fvprc |
⊢ ( ¬ 𝑉 ∈ V → ( Pairs ‘ 𝑉 ) = ∅ ) |
33 |
32
|
sseq2d |
⊢ ( ¬ 𝑉 ∈ V → ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ↔ 𝑃 ⊆ ∅ ) ) |
34 |
|
ss0b |
⊢ ( 𝑃 ⊆ ∅ ↔ 𝑃 = ∅ ) |
35 |
33 34
|
bitrdi |
⊢ ( ¬ 𝑉 ∈ V → ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ↔ 𝑃 = ∅ ) ) |
36 |
|
rex0 |
⊢ ¬ ∃ 𝑐 ∈ ∅ 𝑐 = { 𝑥 , 𝑦 } |
37 |
|
rexeq |
⊢ ( 𝑃 = ∅ → ( ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ↔ ∃ 𝑐 ∈ ∅ 𝑐 = { 𝑥 , 𝑦 } ) ) |
38 |
36 37
|
mtbiri |
⊢ ( 𝑃 = ∅ → ¬ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) |
39 |
38
|
alrimivv |
⊢ ( 𝑃 = ∅ → ∀ 𝑥 ∀ 𝑦 ¬ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) |
40 |
|
opab0 |
⊢ ( { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } = ∅ ↔ ∀ 𝑥 ∀ 𝑦 ¬ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } ) |
41 |
39 40
|
sylibr |
⊢ ( 𝑃 = ∅ → { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } = ∅ ) |
42 |
|
0elpw |
⊢ ∅ ∈ 𝒫 ( 𝑉 × 𝑉 ) |
43 |
41 42
|
eqeltrdi |
⊢ ( 𝑃 = ∅ → { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } ∈ 𝒫 ( 𝑉 × 𝑉 ) ) |
44 |
35 43
|
syl6bi |
⊢ ( ¬ 𝑉 ∈ V → ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } ∈ 𝒫 ( 𝑉 × 𝑉 ) ) ) |
45 |
31 44
|
pm2.61i |
⊢ ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) → { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑃 𝑐 = { 𝑥 , 𝑦 } } ∈ 𝒫 ( 𝑉 × 𝑉 ) ) |