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Theorem prcom 3669
Description: Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
prcom {𝐴, 𝐵} = {𝐵, 𝐴}

Proof of Theorem prcom
StepHypRef Expression
1 uncom 3280 . 2 ({𝐴} ∪ {𝐵}) = ({𝐵} ∪ {𝐴})
2 df-pr 3600 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3 df-pr 3600 . 2 {𝐵, 𝐴} = ({𝐵} ∪ {𝐴})
41, 2, 33eqtr4i 2208 1 {𝐴, 𝐵} = {𝐵, 𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1353  cun 3128  {csn 3593  {cpr 3594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-pr 3600
This theorem is referenced by:  preq2  3671  tpcoma  3687  tpidm23  3694  prid2g  3698  prid2  3700  prprc2  3702  difprsn2  3733  preqr2g  3768  preqr2  3770  preq12b  3771  fvpr2  5722  fvpr2g  5724  en2other2  7195  maxcom  11212  mincom  11237  xrmax2sup  11262  xrmaxltsup  11266  xrmaxadd  11269  xrbdtri  11284  qtopbasss  14024
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