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Mirrors > Home > MPE Home > Th. List > otthg | Structured version Visualization version GIF version |
Description: Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.) |
Ref | Expression |
---|---|
otthg | ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (〈𝐴, 𝐵, 𝐶〉 = 〈𝐷, 𝐸, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 4536 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | df-ot 4536 | . . 3 ⊢ 〈𝐷, 𝐸, 𝐹〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 | |
3 | 1, 2 | eqeq12i 2751 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 = 〈𝐷, 𝐸, 𝐹〉 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉) |
4 | opex 5333 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
5 | opthg 5346 | . . . . 5 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹))) | |
6 | 4, 5 | mpan 690 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹))) |
7 | opthg 5346 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸))) | |
8 | 7 | anbi1d 633 | . . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹))) |
9 | df-3an 1091 | . . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹)) | |
10 | 8, 9 | bitr4di 292 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
11 | 6, 10 | sylan9bbr 514 | . . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
12 | 11 | 3impa 1112 | . 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
13 | 3, 12 | syl5bb 286 | 1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (〈𝐴, 𝐵, 𝐶〉 = 〈𝐷, 𝐸, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 Vcvv 3398 〈cop 4533 〈cotp 4535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-ot 4536 |
This theorem is referenced by: otsndisj 5387 otiunsndisj 5388 otiunsndisjX 44386 |
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