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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 3778 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | eqeq1d 2186 | . . 3 ⊢ (𝑥 = 𝐴 → (〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉)) |
3 | eqeq1 2184 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ↔ 𝐴 = 𝐶)) | |
4 | 3 | anbi1d 465 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷))) |
5 | 2, 4 | bibi12d 235 | . 2 ⊢ (𝑥 = 𝐴 → ((〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)))) |
6 | opeq2 3779 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
7 | 6 | eqeq1d 2186 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉)) |
8 | eqeq1 2184 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐷 ↔ 𝐵 = 𝐷)) | |
9 | 8 | anbi2d 464 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
10 | 7, 9 | bibi12d 235 | . 2 ⊢ (𝑦 = 𝐵 → ((〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))) |
11 | vex 2740 | . . 3 ⊢ 𝑥 ∈ V | |
12 | vex 2740 | . . 3 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | opth 4237 | . 2 ⊢ (〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) |
14 | 5, 10, 13 | vtocl2g 2801 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 〈cop 3595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 |
This theorem is referenced by: opthg2 4239 xpopth 6176 eqop 6177 inl11 7063 preqlu 7470 cauappcvgprlemladd 7656 elrealeu 7827 qnumdenbi 12186 crth 12218 imasaddfnlemg 12729 |
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