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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 4377 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | df-ot 4377 | . . 3 ⊢ 〈𝐷, 𝐸, 𝐹〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 | |
3 | 1, 2 | eqeq12i 2813 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 = 〈𝐷, 𝐸, 𝐹〉 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉) |
4 | opex 5123 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
5 | opthg 5136 | . . . . 5 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹))) | |
6 | 4, 5 | mpan 682 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹))) |
7 | opthg 5136 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸))) | |
8 | 7 | anbi1d 624 | . . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹))) |
9 | df-3an 1110 | . . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹)) | |
10 | 8, 9 | syl6bbr 281 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
11 | 6, 10 | sylan9bbr 507 | . . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
12 | 11 | 3impa 1137 | . 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
13 | 3, 12 | syl5bb 275 | 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 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 Vcvv 3385 〈cop 4374 〈cotp 4376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-ot 4377 |
This theorem is referenced by: otsndisj 5175 otiunsndisj 5176 otiunsndisjX 42134 |
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