Theorem elpr 3715

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 3714	. 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 2205   _Vcvv 2815   {cpr 3695

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701

This theorem is referenced by:  prmg  3819  difprsnss  3837  preqr1  3877  preq12b  3879  prel12  3880  pwprss  3915  pwtpss  3916  unipr  3933  intpr  3986  zfpair2  4328  elop  4352  ordtri2or2exmidlem  4653  onsucelsucexmidlem  4656  en2lp  4681  reg3exmidlemwe  4706  xpsspw  4867  acexmidlem2  6055  2oconcl  6685  exmidpw  7181  exmidpweq  7182  renfdisj  8349  fzpr  10433  maxabslemval  11918  xrmaxiflemval  11960  isprm2  12839  2lgslem4  16102  structiedg0val  16161  bj-zfpair2  16806  ss1oel2o  16887