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Theorem prcom 3647
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 3262 . 2 ({𝐴} ∪ {𝐵}) = ({𝐵} ∪ {𝐴})
2 df-pr 3578 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3 df-pr 3578 . 2 {𝐵, 𝐴} = ({𝐵} ∪ {𝐴})
41, 2, 33eqtr4i 2195 1 {𝐴, 𝐵} = {𝐵, 𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1342  cun 3110  {csn 3571  {cpr 3572
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-un 3116  df-pr 3578
This theorem is referenced by:  preq2  3649  tpcoma  3665  tpidm23  3672  prid2g  3676  prid2  3678  prprc2  3680  difprsn2  3708  preqr2g  3742  preqr2  3744  preq12b  3745  fvpr2  5685  fvpr2g  5687  en2other2  7144  maxcom  11135  mincom  11160  xrmax2sup  11185  xrmaxltsup  11189  xrmaxadd  11192  xrbdtri  11207  qtopbasss  13088
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