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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 6213 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) | |
2 | df-br 3977 | . . 3 ⊢ (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹) | |
3 | df-br 3977 | . . 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 3580 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
6 | 5 | eleq1i 2230 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ tpos 𝐹) |
7 | df-ot 3580 | . . 3 ⊢ 〈𝐵, 𝐴, 𝐶〉 = 〈〈𝐵, 𝐴〉, 𝐶〉 | |
8 | 7 | eleq1i 2230 | . 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 967 ∈ wcel 2135 〈cop 3573 〈cotp 3574 class class class wbr 3976 tpos ctpos 6203 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-ot 3580 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 df-tpos 6204 |
This theorem is referenced by: (None) |
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