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Mirrors > Home > MPE Home > Th. List > tprot | Structured version Visualization version 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 1089 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)) | |
2 | 1 | abbii 2863 | . 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)} |
3 | dftp2 4587 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} | |
4 | dftp2 4587 | . 2 ⊢ {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)} | |
5 | 2, 3, 4 | 3eqtr4i 2831 | 1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1083 = wceq 1538 {cab 2776 {ctp 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-sn 4526 df-pr 4528 df-tp 4530 |
This theorem is referenced by: tpcomb 4647 tpass 4648 tpidm13 4652 tpidm23 4653 tpprceq3 4697 fvtp2 6935 fvtp3 6936 fvtp2g 6938 fvtp3g 6939 f13dfv 7009 en3lplem2 9060 estrres 17381 nb3grprlem2 27171 nb3grpr 27172 nb3grpr2 27173 nb3gr2nb 27174 cplgr3v 27225 frgr3v 28060 1to3vfriswmgr 28065 dvh4dimN 38743 en3lplem2VD 41550 |
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