Proof of Theorem ssprsseq
Step | Hyp | Ref
| Expression |
1 | | ssprss 4296 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))) |
2 | 1 | 3adant3 1074 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))) |
3 | | eqtr3 2631 |
. . . . . . . . 9
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵) |
4 | | eqneqall 2793 |
. . . . . . . . 9
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷})) |
5 | 3, 4 | syl 17 |
. . . . . . . 8
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷})) |
6 | 5 | com12 32 |
. . . . . . 7
⊢ (𝐴 ≠ 𝐵 → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})) |
7 | 6 | 3ad2ant3 1077 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})) |
8 | 7 | com12 32 |
. . . . 5
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷})) |
9 | | preq12 4214 |
. . . . . . 7
⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶}) |
10 | | prcom 4211 |
. . . . . . 7
⊢ {𝐷, 𝐶} = {𝐶, 𝐷} |
11 | 9, 10 | syl6eq 2660 |
. . . . . 6
⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷}) |
12 | 11 | a1d 25 |
. . . . 5
⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷})) |
13 | | preq12 4214 |
. . . . . 6
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) |
14 | 13 | a1d 25 |
. . . . 5
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷})) |
15 | | eqtr3 2631 |
. . . . . . . . 9
⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → 𝐴 = 𝐵) |
16 | 15, 4 | syl 17 |
. . . . . . . 8
⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} = {𝐶, 𝐷})) |
17 | 16 | com12 32 |
. . . . . . 7
⊢ (𝐴 ≠ 𝐵 → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})) |
18 | 17 | 3ad2ant3 1077 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})) |
19 | 18 | com12 32 |
. . . . 5
⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐷) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷})) |
20 | 8, 12, 14, 19 | ccase 984 |
. . . 4
⊢ (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} = {𝐶, 𝐷})) |
21 | 20 | com12 32 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)) → {𝐴, 𝐵} = {𝐶, 𝐷})) |
22 | 2, 21 | sylbid 229 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} → {𝐴, 𝐵} = {𝐶, 𝐷})) |
23 | | eqimss 3620 |
. 2
⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → {𝐴, 𝐵} ⊆ {𝐶, 𝐷}) |
24 | 22, 23 | impbid1 214 |
1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷})) |