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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 1088 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)) | |
2 | 1 | abbii 2888 | . 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)} |
3 | dftp2 4629 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} | |
4 | dftp2 4629 | . 2 ⊢ {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)} | |
5 | 2, 3, 4 | 3eqtr4i 2856 | 1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1082 = wceq 1537 {cab 2801 {ctp 4573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-un 3943 df-sn 4570 df-pr 4572 df-tp 4574 |
This theorem is referenced by: tpcomb 4689 tpass 4690 tpidm13 4694 tpidm23 4695 tpprceq3 4739 fvtp2 6960 fvtp3 6961 fvtp2g 6963 fvtp3g 6964 f13dfv 7033 en3lplem2 9078 estrres 17391 nb3grprlem2 27165 nb3grpr 27166 nb3grpr2 27167 nb3gr2nb 27168 cplgr3v 27219 frgr3v 28056 1to3vfriswmgr 28061 dvh4dimN 38585 en3lplem2VD 41185 |
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