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Mirrors > Home > ILE Home > Th. List > tprot | GIF version |
Description: Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.) |
Ref | Expression |
---|---|
tprot | ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3orrot 968 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)) | |
2 | 1 | abbii 2255 | . 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)} |
3 | dftp2 3572 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} | |
4 | dftp2 3572 | . 2 ⊢ {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)} | |
5 | 2, 3, 4 | 3eqtr4i 2170 | 1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} |
Colors of variables: wff set class |
Syntax hints: ∨ w3o 961 = wceq 1331 {cab 2125 {ctp 3529 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-tp 3535 |
This theorem is referenced by: tpcomb 3618 tpass 3619 tpidm13 3623 tpidm23 3624 prsstp23 3675 fvtp2g 5629 fvtp3g 5630 fvtp2 5632 fvtp3 5633 |
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