Step |
Hyp |
Ref |
Expression |
1 |
|
sprel |
⊢ ( { 𝑋 , 𝑌 } ∈ ( Pairs ‘ 𝑉 ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 { 𝑋 , 𝑌 } = { 𝑎 , 𝑏 } ) |
2 |
|
preq12bg |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( { 𝑋 , 𝑌 } = { 𝑎 , 𝑏 } ↔ ( ( 𝑋 = 𝑎 ∧ 𝑌 = 𝑏 ) ∨ ( 𝑋 = 𝑏 ∧ 𝑌 = 𝑎 ) ) ) ) |
3 |
|
eleq1 |
⊢ ( 𝑎 = 𝑋 → ( 𝑎 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉 ) ) |
4 |
3
|
eqcoms |
⊢ ( 𝑋 = 𝑎 → ( 𝑎 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉 ) ) |
5 |
|
eleq1 |
⊢ ( 𝑏 = 𝑌 → ( 𝑏 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉 ) ) |
6 |
5
|
eqcoms |
⊢ ( 𝑌 = 𝑏 → ( 𝑏 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉 ) ) |
7 |
4 6
|
bi2anan9 |
⊢ ( ( 𝑋 = 𝑎 ∧ 𝑌 = 𝑏 ) → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ↔ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
8 |
7
|
biimpd |
⊢ ( ( 𝑋 = 𝑎 ∧ 𝑌 = 𝑏 ) → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
9 |
|
eleq1 |
⊢ ( 𝑏 = 𝑋 → ( 𝑏 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉 ) ) |
10 |
9
|
eqcoms |
⊢ ( 𝑋 = 𝑏 → ( 𝑏 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉 ) ) |
11 |
|
eleq1 |
⊢ ( 𝑎 = 𝑌 → ( 𝑎 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉 ) ) |
12 |
11
|
eqcoms |
⊢ ( 𝑌 = 𝑎 → ( 𝑎 ∈ 𝑉 ↔ 𝑌 ∈ 𝑉 ) ) |
13 |
10 12
|
bi2anan9 |
⊢ ( ( 𝑋 = 𝑏 ∧ 𝑌 = 𝑎 ) → ( ( 𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ) ↔ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
14 |
13
|
biimpd |
⊢ ( ( 𝑋 = 𝑏 ∧ 𝑌 = 𝑎 ) → ( ( 𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
15 |
14
|
ancomsd |
⊢ ( ( 𝑋 = 𝑏 ∧ 𝑌 = 𝑎 ) → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
16 |
8 15
|
jaoi |
⊢ ( ( ( 𝑋 = 𝑎 ∧ 𝑌 = 𝑏 ) ∨ ( 𝑋 = 𝑏 ∧ 𝑌 = 𝑎 ) ) → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
17 |
16
|
com12 |
⊢ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( ( ( 𝑋 = 𝑎 ∧ 𝑌 = 𝑏 ) ∨ ( 𝑋 = 𝑏 ∧ 𝑌 = 𝑎 ) ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
18 |
17
|
adantl |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ( ( 𝑋 = 𝑎 ∧ 𝑌 = 𝑏 ) ∨ ( 𝑋 = 𝑏 ∧ 𝑌 = 𝑎 ) ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
19 |
2 18
|
sylbid |
⊢ ( ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( { 𝑋 , 𝑌 } = { 𝑎 , 𝑏 } → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
20 |
19
|
expcom |
⊢ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) → ( { 𝑋 , 𝑌 } = { 𝑎 , 𝑏 } → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) ) |
21 |
20
|
com23 |
⊢ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( { 𝑋 , 𝑌 } = { 𝑎 , 𝑏 } → ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) ) |
22 |
21
|
rexlimivv |
⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 { 𝑋 , 𝑌 } = { 𝑎 , 𝑏 } → ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
23 |
1 22
|
syl |
⊢ ( { 𝑋 , 𝑌 } ∈ ( Pairs ‘ 𝑉 ) → ( ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
24 |
23
|
imp |
⊢ ( ( { 𝑋 , 𝑌 } ∈ ( Pairs ‘ 𝑉 ) ∧ ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) |