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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 7904 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) | |
2 | df-br 5070 | . . 3 ⊢ (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹) | |
3 | df-br 5070 | . . 3 ⊢ (〈𝐵, 𝐴〉𝐹𝐶 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹) | |
4 | 1, 2, 3 | 3bitr3g 315 | . 2 ⊢ (𝐶 ∈ 𝑉 → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹)) |
5 | df-ot 4579 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
6 | 5 | eleq1i 2906 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹) |
7 | df-ot 4579 | . . 3 ⊢ 〈𝐵, 𝐴, 𝐶〉 = 〈〈𝐵, 𝐴〉, 𝐶〉 | |
8 | 7 | eleq1i 2906 | . 2 ⊢ (〈𝐵, 𝐴, 𝐶〉 ∈ 𝐹 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹) |
9 | 4, 6, 8 | 3bitr4g 316 | 1 ⊢ (𝐶 ∈ 𝑉 → (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈𝐵, 𝐴, 𝐶〉 ∈ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2113 〈cop 4576 〈cotp 4578 class class class wbr 5069 tpos ctpos 7894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-ot 4579 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-fv 6366 df-tpos 7895 |
This theorem is referenced by: (None) |
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