Theorem elpr 3694

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 3693	. 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 105   \/ wo 716   = wceq 1398   e. wcel 2202   _Vcvv 2803   {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680

This theorem is referenced by:  prmg  3798  difprsnss  3816  preqr1  3856  preq12b  3858  prel12  3859  pwprss  3894  pwtpss  3895  unipr  3912  intpr  3965  zfpair2  4306  elop  4329  ordtri2or2exmidlem  4630  onsucelsucexmidlem  4633  en2lp  4658  reg3exmidlemwe  4683  xpsspw  4844  acexmidlem2  6025  2oconcl  6650  exmidpw  7143  exmidpweq  7144  renfdisj  8298  fzpr  10374  maxabslemval  11848  xrmaxiflemval  11890  isprm2  12769  2lgslem4  15922  structiedg0val  15981  bj-zfpair2  16626  ss1oel2o  16707