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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 8170 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)) | |
2 | df-br 5110 | . . 3 ⊢ (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ tpos 𝐹) | |
3 | df-br 5110 | . . 3 ⊢ (⟨𝐵, 𝐴⟩𝐹𝐶 ↔ ⟨⟨𝐵, 𝐴⟩, 𝐶⟩ ∈ 𝐹) | |
4 | 1, 2, 3 | 3bitr3g 313 | . 2 ⊢ (𝐶 ∈ 𝑉 → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ tpos 𝐹 ↔ ⟨⟨𝐵, 𝐴⟩, 𝐶⟩ ∈ 𝐹)) |
5 | df-ot 4599 | . . 3 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ | |
6 | 5 | eleq1i 2825 | . 2 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ∈ tpos 𝐹 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ tpos 𝐹) |
7 | df-ot 4599 | . . 3 ⊢ ⟨𝐵, 𝐴, 𝐶⟩ = ⟨⟨𝐵, 𝐴⟩, 𝐶⟩ | |
8 | 7 | eleq1i 2825 | . 2 ⊢ (⟨𝐵, 𝐴, 𝐶⟩ ∈ 𝐹 ↔ ⟨⟨𝐵, 𝐴⟩, 𝐶⟩ ∈ 𝐹) |
9 | 4, 6, 8 | 3bitr4g 314 | 1 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵, 𝐶⟩ ∈ tpos 𝐹 ↔ ⟨𝐵, 𝐴, 𝐶⟩ ∈ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 ⟨cop 4596 ⟨cotp 4598 class class class wbr 5109 tpos ctpos 8160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-ot 4599 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-fv 6508 df-tpos 8161 |
This theorem is referenced by: (None) |
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