Theorem elpr 3553

Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Hypothesis

Ref	Expression
elpr.1	|- A e. _V

Assertion

Ref	Expression
elpr	|- ( A e. { B , C } <-> ( A = B \/ A = C ) )

Proof of Theorem elpr

Step	Hyp	Ref	Expression
1		elpr.1	. 2 |- A e. _V
2		elprg 3552	. 2 |- ( A e. _V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
3	1, 2	ax-mp 5	1 |- ( A e. { B , C } <-> ( A = B \/ A = C ) )

Syntax hints:   <-> wb 104   \/ wo 698   = wceq 1332   e. wcel 1481   _Vcvv 2689   {cpr 3533

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539

This theorem is referenced by:  prmg  3652  difprsnss  3666  preqr1  3703  preq12b  3705  prel12  3706  pwprss  3740  pwtpss  3741  unipr  3758  intpr  3811  zfpair2  4140  elop  4161  ordtri2or2exmidlem  4449  onsucelsucexmidlem  4452  en2lp  4477  reg3exmidlemwe  4501  xpsspw  4659  acexmidlem2  5779  2oconcl  6344  exmidpw  6810  renfdisj  7848  fzpr  9888  maxabslemval  11012  xrmaxiflemval  11051  isprm2  11834  bj-zfpair2  13279  ss1oel2o  13360