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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 4550 | . . 3 ⊢ 〈𝐴, 𝐵, 𝑅〉 = 〈〈𝐴, 𝐵〉, 𝑅〉 | |
2 | df-ot 4550 | . . 3 ⊢ 〈𝐶, 𝐷, 𝑆〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 | |
3 | 1, 2 | eqeq12i 2755 | . 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 5367 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
8 | 3, 7 | bitri 278 | 1 ⊢ (〈𝐴, 𝐵, 𝑅〉 = 〈𝐶, 𝐷, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 Vcvv 3408 〈cop 4547 〈cotp 4549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-ot 4550 |
This theorem is referenced by: euotd 5396 mthmpps 33257 ackval40 45712 |
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