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Mirrors > Home > ILE Home > Th. List > opthpr | GIF version |
Description: A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.) |
Ref | Expression |
---|---|
preq12b.1 | ⊢ A ∈ V |
preq12b.2 | ⊢ B ∈ V |
preq12b.3 | ⊢ 𝐶 ∈ V |
preq12b.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
opthpr | ⊢ (A ≠ 𝐷 → ({A, B} = {𝐶, 𝐷} ↔ (A = 𝐶 ∧ B = 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq12b.1 | . . 3 ⊢ A ∈ V | |
2 | preq12b.2 | . . 3 ⊢ B ∈ V | |
3 | preq12b.3 | . . 3 ⊢ 𝐶 ∈ V | |
4 | preq12b.4 | . . 3 ⊢ 𝐷 ∈ V | |
5 | 1, 2, 3, 4 | preq12b 3532 | . 2 ⊢ ({A, B} = {𝐶, 𝐷} ↔ ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶))) |
6 | idd 21 | . . . 4 ⊢ (A ≠ 𝐷 → ((A = 𝐶 ∧ B = 𝐷) → (A = 𝐶 ∧ B = 𝐷))) | |
7 | df-ne 2203 | . . . . . 6 ⊢ (A ≠ 𝐷 ↔ ¬ A = 𝐷) | |
8 | pm2.21 547 | . . . . . 6 ⊢ (¬ A = 𝐷 → (A = 𝐷 → (B = 𝐶 → (A = 𝐶 ∧ B = 𝐷)))) | |
9 | 7, 8 | sylbi 114 | . . . . 5 ⊢ (A ≠ 𝐷 → (A = 𝐷 → (B = 𝐶 → (A = 𝐶 ∧ B = 𝐷)))) |
10 | 9 | impd 242 | . . . 4 ⊢ (A ≠ 𝐷 → ((A = 𝐷 ∧ B = 𝐶) → (A = 𝐶 ∧ B = 𝐷))) |
11 | 6, 10 | jaod 636 | . . 3 ⊢ (A ≠ 𝐷 → (((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)) → (A = 𝐶 ∧ B = 𝐷))) |
12 | orc 632 | . . 3 ⊢ ((A = 𝐶 ∧ B = 𝐷) → ((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶))) | |
13 | 11, 12 | impbid1 130 | . 2 ⊢ (A ≠ 𝐷 → (((A = 𝐶 ∧ B = 𝐷) ∨ (A = 𝐷 ∧ B = 𝐶)) ↔ (A = 𝐶 ∧ B = 𝐷))) |
14 | 5, 13 | syl5bb 181 | 1 ⊢ (A ≠ 𝐷 → ({A, B} = {𝐶, 𝐷} ↔ (A = 𝐶 ∧ B = 𝐷))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 628 = wceq 1242 ∈ wcel 1390 ≠ wne 2201 Vcvv 2551 {cpr 3368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 |
This theorem is referenced by: (None) |
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