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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 4763 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | eqeq1d 2800 | . . 3 ⊢ (𝑥 = 𝐴 → (〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉)) |
3 | eqeq1 2802 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ↔ 𝐴 = 𝐶)) | |
4 | 3 | anbi1d 632 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷))) |
5 | 2, 4 | bibi12d 349 | . 2 ⊢ (𝑥 = 𝐴 → ((〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)))) |
6 | opeq2 4765 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
7 | 6 | eqeq1d 2800 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉)) |
8 | eqeq1 2802 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐷 ↔ 𝐵 = 𝐷)) | |
9 | 8 | anbi2d 631 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
10 | 7, 9 | bibi12d 349 | . 2 ⊢ (𝑦 = 𝐵 → ((〈𝐴, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))) |
11 | vex 3444 | . . 3 ⊢ 𝑥 ∈ V | |
12 | vex 3444 | . . 3 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | opth 5333 | . 2 ⊢ (〈𝑥, 𝑦〉 = 〈𝐶, 𝐷〉 ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) |
14 | 5, 10, 13 | vtocl2g 3520 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 〈cop 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 |
This theorem is referenced by: opth1g 5335 opthg2 5336 opthneg 5338 otthg 5342 oteqex 5355 s111 13960 embedsetcestrclem 17399 symg2bas 18513 frgpnabllem1 18986 frgpnabllem2 18987 mat1dimbas 21077 linds2eq 30995 goeleq12bg 32709 opideq 35760 dvheveccl 38408 hoidmv1le 43233 oppr 43622 opprb 43623 prproropf1olem4 44023 |
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