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Mirrors > Home > MPE Home > Th. List > ottpos | Structured version Visualization version GIF version |
Description: The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
ottpos | ⊢ (𝐶 ∈ 𝑉 → (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈𝐵, 𝐴, 𝐶〉 ∈ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brtpos 7530 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) | |
2 | df-br 4805 | . . 3 ⊢ (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹) | |
3 | df-br 4805 | . . 3 ⊢ (〈𝐵, 𝐴〉𝐹𝐶 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹) | |
4 | 1, 2, 3 | 3bitr3g 302 | . 2 ⊢ (𝐶 ∈ 𝑉 → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹)) |
5 | df-ot 4330 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
6 | 5 | eleq1i 2830 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹) |
7 | df-ot 4330 | . . 3 ⊢ 〈𝐵, 𝐴, 𝐶〉 = 〈〈𝐵, 𝐴〉, 𝐶〉 | |
8 | 7 | eleq1i 2830 | . 2 ⊢ (〈𝐵, 𝐴, 𝐶〉 ∈ 𝐹 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹) |
9 | 4, 6, 8 | 3bitr4g 303 | 1 ⊢ (𝐶 ∈ 𝑉 → (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈𝐵, 𝐴, 𝐶〉 ∈ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 2139 〈cop 4327 〈cotp 4329 class class class wbr 4804 tpos ctpos 7520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-ot 4330 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-fv 6057 df-tpos 7521 |
This theorem is referenced by: (None) |
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