Step |
Hyp |
Ref |
Expression |
1 |
|
elneq |
⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵 ) |
2 |
1
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷 ) → 𝐴 ≠ 𝐵 ) |
3 |
|
preq12nebg |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵 ) → ( { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ↔ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ) ) ) |
4 |
2 3
|
syld3an3 |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷 ) → ( { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ↔ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ) ) ) |
5 |
|
eleq12 |
⊢ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) → ( 𝐴 ∈ 𝐵 ↔ 𝐷 ∈ 𝐶 ) ) |
6 |
|
elnotel |
⊢ ( 𝐷 ∈ 𝐶 → ¬ 𝐶 ∈ 𝐷 ) |
7 |
6
|
pm2.21d |
⊢ ( 𝐷 ∈ 𝐶 → ( 𝐶 ∈ 𝐷 → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |
8 |
5 7
|
syl6bi |
⊢ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) → ( 𝐴 ∈ 𝐵 → ( 𝐶 ∈ 𝐷 → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) ) |
9 |
8
|
com3l |
⊢ ( 𝐴 ∈ 𝐵 → ( 𝐶 ∈ 𝐷 → ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) ) |
10 |
9
|
a1d |
⊢ ( 𝐴 ∈ 𝐵 → ( 𝐵 ∈ 𝑉 → ( 𝐶 ∈ 𝐷 → ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) ) ) |
11 |
10
|
3imp |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷 ) → ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |
12 |
11
|
com12 |
⊢ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) → ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷 ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |
13 |
12
|
jao1i |
⊢ ( ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ) → ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷 ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |
14 |
13
|
com12 |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷 ) → ( ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |
15 |
4 14
|
sylbid |
⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷 ) → ( { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |
16 |
15
|
imp |
⊢ ( ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷 ) ∧ { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) |