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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 4633 | . . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | 1 | uneq1i 4059 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶}) |
3 | df-tp 4531 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
4 | df-tp 4531 | . 2 ⊢ {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶}) | |
5 | 2, 3, 4 | 3eqtr4i 2772 | 1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∪ cun 3851 {csn 4526 {cpr 4528 {ctp 4530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-v 3402 df-un 3858 df-pr 4529 df-tp 4531 |
This theorem is referenced by: tpcomb 4652 tppreqb 4703 nb3grpr2 27337 nb3gr2nb 27338 frgr3v 28224 3vfriswmgr 28227 1to3vfriswmgr 28229 |
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