Theorem elpr 3664

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 3663	. 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 710   = wceq 1373   e. wcel 2178   _Vcvv 2776   {cpr 3644

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650

This theorem is referenced by:  prmg  3765  difprsnss  3782  preqr1  3822  preq12b  3824  prel12  3825  pwprss  3860  pwtpss  3861  unipr  3878  intpr  3931  zfpair2  4270  elop  4293  ordtri2or2exmidlem  4592  onsucelsucexmidlem  4595  en2lp  4620  reg3exmidlemwe  4645  xpsspw  4805  acexmidlem2  5964  2oconcl  6548  exmidpw  7031  exmidpweq  7032  renfdisj  8167  fzpr  10234  maxabslemval  11634  xrmaxiflemval  11676  isprm2  12554  2lgslem4  15695  structiedg0val  15754  bj-zfpair2  16045  ss1oel2o  16127