ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prcom GIF version

Theorem prcom 3668
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 3279 . 2 ({𝐴} ∪ {𝐵}) = ({𝐵} ∪ {𝐴})
2 df-pr 3599 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3 df-pr 3599 . 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 3127  {csn 3592  {cpr 3593
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 2739  df-un 3133  df-pr 3599
This theorem is referenced by:  preq2  3670  tpcoma  3686  tpidm23  3693  prid2g  3697  prid2  3699  prprc2  3701  difprsn2  3732  preqr2g  3767  preqr2  3769  preq12b  3770  fvpr2  5721  fvpr2g  5723  en2other2  7194  maxcom  11211  mincom  11236  xrmax2sup  11261  xrmaxltsup  11265  xrmaxadd  11268  xrbdtri  11283  qtopbasss  13991
  Copyright terms: Public domain W3C validator