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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 4640 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | df-ot 4640 | . . 3 ⊢ 〈𝐷, 𝐸, 𝐹〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 | |
3 | 1, 2 | eqeq12i 2753 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 = 〈𝐷, 𝐸, 𝐹〉 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉) |
4 | opex 5475 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
5 | opthg 5488 | . . . . 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 5488 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸))) | |
8 | 7 | anbi1d 631 | . . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹))) |
9 | df-3an 1088 | . . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹)) | |
10 | 8, 9 | bitr4di 289 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
11 | 6, 10 | sylan9bbr 510 | . . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
12 | 11 | 3impa 1109 | . 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
13 | 3, 12 | bitrid 283 | 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 〈cotp 4639 |
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 df-ot 4640 |
This theorem is referenced by: otsndisj 5529 otiunsndisj 5530 otiunsndisjX 47229 |
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