| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 4732 | . . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
| 2 | 1 | uneq1i 4164 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶}) |
| 3 | df-tp 4631 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
| 4 | df-tp 4631 | . 2 ⊢ {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶}) | |
| 5 | 2, 3, 4 | 3eqtr4i 2775 | 1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cun 3949 {csn 4626 {cpr 4628 {ctp 4630 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-pr 4629 df-tp 4631 |
| This theorem is referenced by: tpcomb 4751 tppreqb 4805 nb3grpr2 29400 nb3gr2nb 29401 frgr3v 30294 3vfriswmgr 30297 1to3vfriswmgr 30299 |
| Copyright terms: Public domain | W3C validator |