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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 4211 | . . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | 1 | uneq1i 3725 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶}) |
3 | df-tp 4130 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
4 | df-tp 4130 | . 2 ⊢ {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶}) | |
5 | 2, 3, 4 | 3eqtr4i 2642 | 1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∪ cun 3538 {csn 4125 {cpr 4127 {ctp 4129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-pr 4128 df-tp 4130 |
This theorem is referenced by: tpcomb 4230 tppreqb 4277 nb3grapr2 25983 nb3gra2nb 25984 frgra3v 26529 3vfriswmgra 26532 1to3vfriswmgra 26534 nb3grpr2 40611 nb3gr2nb 40612 frgr3v 41445 3vfriswmgr 41448 1to3vfriswmgr 41450 |
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