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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 4340 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | eqeq1d 2612 | . . 3 ⊢ (𝑥 = 𝐴 → (〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉)) |
3 | eqeq1 2614 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ↔ 𝐴 = 𝐶)) | |
4 | 3 | anbi1d 737 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷))) |
5 | 2, 4 | bibi12d 334 | . 2 ⊢ (𝑥 = 𝐴 → ((〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)))) |
6 | opeq2 4341 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
7 | 6 | eqeq1d 2612 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉)) |
8 | eqeq1 2614 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐷 ↔ 𝐵 = 𝐷)) | |
9 | 8 | anbi2d 736 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
10 | 7, 9 | bibi12d 334 | . 2 ⊢ (𝑦 = 𝐵 → ((〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))) |
11 | vex 3176 | . . 3 ⊢ 𝑥 ∈ V | |
12 | vex 3176 | . . 3 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | opth 4871 | . 2 ⊢ (〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) |
14 | 5, 10, 13 | vtocl2g 3243 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 |
This theorem is referenced by: opth1g 4873 opthg2 4874 opthneg 4876 otthg 4880 oteqex 4889 s111 13248 embedsetcestrclem 16620 symg2bas 17641 frgpnabllem1 18099 frgpnabllem2 18100 mat1dimbas 20097 el2wlkonotot0 26399 dvheveccl 35419 hoidmv1le 39484 |
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