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Mirrors > Home > ILE Home > Th. List > opthg | GIF version |
Description: Ordered pair theorem. 𝐶 and 𝐷 are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opthg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3713 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | eqeq1d 2149 | . . 3 ⊢ (𝑥 = 𝐴 → (〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉)) |
3 | eqeq1 2147 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ↔ 𝐴 = 𝐶)) | |
4 | 3 | anbi1d 461 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷))) |
5 | 2, 4 | bibi12d 234 | . 2 ⊢ (𝑥 = 𝐴 → ((〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)))) |
6 | opeq2 3714 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
7 | 6 | eqeq1d 2149 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉)) |
8 | eqeq1 2147 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐷 ↔ 𝐵 = 𝐷)) | |
9 | 8 | anbi2d 460 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
10 | 7, 9 | bibi12d 234 | . 2 ⊢ (𝑦 = 𝐵 → ((〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))) |
11 | vex 2692 | . . 3 ⊢ 𝑥 ∈ V | |
12 | vex 2692 | . . 3 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | opth 4167 | . 2 ⊢ (〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) |
14 | 5, 10, 13 | vtocl2g 2753 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 〈cop 3535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 |
This theorem is referenced by: opthg2 4169 xpopth 6082 eqop 6083 inl11 6958 preqlu 7304 cauappcvgprlemladd 7490 elrealeu 7661 qnumdenbi 11906 crth 11936 |
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