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Mirrors > Home > MPE Home > Th. List > opthg | Structured version Visualization version 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 4878 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | eqeq1d 2737 | . . 3 ⊢ (𝑥 = 𝐴 → (〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉)) |
3 | eqeq1 2739 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ↔ 𝐴 = 𝐶)) | |
4 | 3 | anbi1d 631 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷))) |
5 | 2, 4 | bibi12d 345 | . 2 ⊢ (𝑥 = 𝐴 → ((〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)))) |
6 | opeq2 4879 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
7 | 6 | eqeq1d 2737 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉)) |
8 | eqeq1 2739 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐷 ↔ 𝐵 = 𝐷)) | |
9 | 8 | anbi2d 630 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
10 | 7, 9 | bibi12d 345 | . 2 ⊢ (𝑦 = 𝐵 → ((〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))) |
11 | vex 3482 | . . 3 ⊢ 𝑥 ∈ V | |
12 | vex 3482 | . . 3 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | opth 5487 | . 2 ⊢ (〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) |
14 | 5, 10, 13 | vtocl2g 3574 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 〈cop 4637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 |
This theorem is referenced by: opth1g 5489 opthg2 5490 opthneg 5492 otthg 5496 oteqex 5510 s111 14650 embedsetcestrclem 18213 symg2bas 19425 frgpnabllem1 19906 frgpnabllem2 19907 mat1dimbas 22494 linds2eq 33389 goeleq12bg 35334 opideq 38325 dvheveccl 41095 hoidmv1le 46550 oppr 46980 opprb 46981 fsetsnf1 47002 prproropf1olem4 47431 |
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