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Mirrors > Home > MPE Home > Th. List > tpcoma | Structured version Visualization version GIF version |
Description: Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015.) |
Ref | Expression |
---|---|
tpcoma | ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 4628 | . . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | 1 | uneq1i 4086 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶}) |
3 | df-tp 4530 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
4 | df-tp 4530 | . 2 ⊢ {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶}) | |
5 | 2, 3, 4 | 3eqtr4i 2831 | 1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∪ cun 3879 {csn 4525 {cpr 4527 {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-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-pr 4528 df-tp 4530 |
This theorem is referenced by: tpcomb 4647 tppreqb 4698 nb3grpr2 27173 nb3gr2nb 27174 frgr3v 28060 3vfriswmgr 28063 1to3vfriswmgr 28065 |
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