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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 1092 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)) | |
2 | 1 | abbii 2802 | . 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)} |
3 | dftp2 4693 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} | |
4 | dftp2 4693 | . 2 ⊢ {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)} | |
5 | 2, 3, 4 | 3eqtr4i 2770 | 1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1086 = wceq 1541 {cab 2709 {ctp 4632 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-un 3953 df-sn 4629 df-pr 4631 df-tp 4633 |
This theorem is referenced by: tpcomb 4755 tpass 4756 tpidm13 4760 tpidm23 4761 tpprceq3 4807 fvtp2 7199 fvtp3 7200 fvtp2g 7202 fvtp3g 7203 f13dfv 7274 en3lplem2 9610 estrres 18095 nb3grprlem2 28893 nb3grpr 28894 nb3grpr2 28895 nb3gr2nb 28896 cplgr3v 28947 frgr3v 29783 1to3vfriswmgr 29788 dvh4dimN 40621 en3lplem2VD 43907 |
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