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Mirrors > Home > ILE Home > Th. List > ottposg | 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 |
---|---|
ottposg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈𝐵, 𝐴, 𝐶〉 ∈ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brtposg 6144 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) | |
2 | df-br 3925 | . . 3 ⊢ (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹) | |
3 | df-br 3925 | . . 3 ⊢ (〈𝐵, 𝐴〉𝐹𝐶 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹) | |
4 | 1, 2, 3 | 3bitr3g 221 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹)) |
5 | df-ot 3532 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
6 | 5 | eleq1i 2203 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹) |
7 | df-ot 3532 | . . 3 ⊢ 〈𝐵, 𝐴, 𝐶〉 = 〈〈𝐵, 𝐴〉, 𝐶〉 | |
8 | 7 | eleq1i 2203 | . 2 ⊢ (〈𝐵, 𝐴, 𝐶〉 ∈ 𝐹 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹) |
9 | 4, 6, 8 | 3bitr4g 222 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈𝐵, 𝐴, 𝐶〉 ∈ 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 962 ∈ wcel 1480 〈cop 3525 〈cotp 3526 class class class wbr 3924 tpos ctpos 6134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-ot 3532 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 df-tpos 6135 |
This theorem is referenced by: (None) |
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