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Mirrors > Home > MPE Home > Th. List > otth2 | Structured version Visualization version GIF version |
Description: Ordered triple theorem, with triple expressed with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
otth.1 | ⊢ 𝐴 ∈ V |
otth.2 | ⊢ 𝐵 ∈ V |
otth.3 | ⊢ 𝑅 ∈ V |
Ref | Expression |
---|---|
otth2 | ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | otth.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | otth.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | opth 5333 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
4 | 3 | anbi1i 626 | . 2 ⊢ ((〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆)) |
5 | opex 5321 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
6 | otth.3 | . . 3 ⊢ 𝑅 ∈ V | |
7 | 5, 6 | opth 5333 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ∧ 𝑅 = 𝑆)) |
8 | df-3an 1086 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆)) | |
9 | 4, 7, 8 | 3bitr4i 306 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 |
This theorem is referenced by: otth 5341 oprabidw 7166 oprabid 7167 eloprabga 7240 |
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