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Theorem prcom 3473
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 3114 . 2  |-  ( { A }  u.  { B } )  =  ( { B }  u.  { A } )
2 df-pr 3409 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
3 df-pr 3409 . 2  |-  { B ,  A }  =  ( { B }  u.  { A } )
41, 2, 33eqtr4i 2086 1  |-  { A ,  B }  =  { B ,  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1259    u. cun 2942   {csn 3402   {cpr 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-pr 3409
This theorem is referenced by:  preq2  3475  tpcoma  3491  tpidm23  3498  prid2g  3502  prid2  3504  prprc2  3506  difprsn2  3531  preqr2g  3565  preqr2  3567  preq12b  3568  fvpr2  5393  fvpr2g  5395
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