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Mirrors > Home > MPE Home > Th. List > otth | Structured version Visualization version GIF version |
Description: Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
otth.1 | ⊢ 𝐴 ∈ V |
otth.2 | ⊢ 𝐵 ∈ V |
otth.3 | ⊢ 𝑅 ∈ V |
Ref | Expression |
---|---|
otth | ⊢ (〈𝐴, 𝐵, 𝑅〉 = 〈𝐶, 𝐷, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 4566 | . . 3 ⊢ 〈𝐴, 𝐵, 𝑅〉 = 〈〈𝐴, 𝐵〉, 𝑅〉 | |
2 | df-ot 4566 | . . 3 ⊢ 〈𝐶, 𝐷, 𝑆〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 | |
3 | 1, 2 | eqeq12i 2833 | . 2 ⊢ (〈𝐴, 𝐵, 𝑅〉 = 〈𝐶, 𝐷, 𝑆〉 ↔ 〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉) |
4 | otth.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | otth.2 | . . 3 ⊢ 𝐵 ∈ V | |
6 | otth.3 | . . 3 ⊢ 𝑅 ∈ V | |
7 | 4, 5, 6 | otth2 5366 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
8 | 3, 7 | bitri 276 | 1 ⊢ (〈𝐴, 𝐵, 𝑅〉 = 〈𝐶, 𝐷, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 〈cop 4563 〈cotp 4565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-ot 4566 |
This theorem is referenced by: euotd 5394 mthmpps 32726 |
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