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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 4186 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | df-ot 4186 | . . 3 ⊢ 〈𝐷, 𝐸, 𝐹〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 | |
3 | 1, 2 | eqeq12i 2636 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 = 〈𝐷, 𝐸, 𝐹〉 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉) |
4 | opex 4932 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
5 | opthg 4946 | . . . . 5 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹))) | |
6 | 4, 5 | mpan 706 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹))) |
7 | opthg 4946 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸))) | |
8 | 7 | anbi1d 741 | . . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹))) |
9 | df-3an 1039 | . . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹)) | |
10 | 8, 9 | syl6bbr 278 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
11 | 6, 10 | sylan9bbr 737 | . . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
12 | 11 | 3impa 1259 | . 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) |
13 | 3, 12 | syl5bb 272 | 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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 〈cotp 4185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 |
This theorem is referenced by: otsndisj 4979 otiunsndisj 4980 otiunsndisjX 41298 |
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