Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > tpcoma | GIF version |
Description: Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.) |
Ref | Expression |
---|---|
tpcoma | ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 3599 | . . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | 1 | uneq1i 3226 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶}) |
3 | df-tp 3535 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
4 | df-tp 3535 | . 2 ⊢ {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶}) | |
5 | 2, 3, 4 | 3eqtr4i 2170 | 1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∪ cun 3069 {csn 3527 {cpr 3528 {ctp 3529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-pr 3534 df-tp 3535 |
This theorem is referenced by: tpcomb 3618 |
Copyright terms: Public domain | W3C validator |