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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 4640 | . . 3 ⊢ 〈𝐴, 𝐵, 𝑅〉 = 〈〈𝐴, 𝐵〉, 𝑅〉 | |
2 | df-ot 4640 | . . 3 ⊢ 〈𝐶, 𝐷, 𝑆〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 | |
3 | 1, 2 | eqeq12i 2753 | . 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 5494 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
8 | 3, 7 | bitri 275 | 1 ⊢ (〈𝐴, 𝐵, 𝑅〉 = 〈𝐶, 𝐷, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ 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: otthne 5497 euotd 5523 mthmpps 35567 ackval40 48543 |
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