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Theorem prcom 3476
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 3117 . 2 ({𝐴} ∪ {𝐵}) = ({𝐵} ∪ {𝐴})
2 df-pr 3413 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3 df-pr 3413 . 2 {𝐵, 𝐴} = ({𝐵} ∪ {𝐴})
41, 2, 33eqtr4i 2112 1 {𝐴, 𝐵} = {𝐵, 𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1285  cun 2972  {csn 3406  {cpr 3407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-pr 3413
This theorem is referenced by:  preq2  3478  tpcoma  3494  tpidm23  3501  prid2g  3505  prid2  3507  prprc2  3509  difprsn2  3534  preqr2g  3567  preqr2  3569  preq12b  3570  fvpr2  5398  fvpr2g  5400  en2other2  6522  maxcom  10227  mincom  10249
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