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Theorem prcom 3798
 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 3408 . 2 ({A} ∪ {B}) = ({B} ∪ {A})
2 df-pr 3742 . 2 {A, B} = ({A} ∪ {B})
3 df-pr 3742 . 2 {B, A} = ({B} ∪ {A})
41, 2, 33eqtr4i 2383 1 {A, B} = {B, A}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∪ cun 3207  {csn 3737  {cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-pr 3742 This theorem is referenced by:  preq2  3800  tpcoma  3816  tpidm23  3823  prid2g  3826  prid2  3828  difprsn2  3848  snprss2  4121  prprc1  4123  preqr2  4125  preq12b  4127  fvpr2  5450
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