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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 6248 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) | |
2 | df-br 4001 | . . 3 ⊢ (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹) | |
3 | df-br 4001 | . . 3 ⊢ (〈𝐵, 𝐴〉𝐹𝐶 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹) | |
4 | 1, 2, 3 | 3bitr3g 222 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹)) |
5 | df-ot 3601 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
6 | 5 | eleq1i 2243 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹) |
7 | df-ot 3601 | . . 3 ⊢ 〈𝐵, 𝐴, 𝐶〉 = 〈〈𝐵, 𝐴〉, 𝐶〉 | |
8 | 7 | eleq1i 2243 | . 2 ⊢ (〈𝐵, 𝐴, 𝐶〉 ∈ 𝐹 ↔ 〈〈𝐵, 𝐴〉, 𝐶〉 ∈ 𝐹) |
9 | 4, 6, 8 | 3bitr4g 223 | 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 105 ∧ w3a 978 ∈ wcel 2148 〈cop 3594 〈cotp 3595 class class class wbr 4000 tpos ctpos 6238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-ot 3601 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-tpos 6239 |
This theorem is referenced by: (None) |
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