Theorem elpr 3654

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 3653	. 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 2176   _Vcvv 2772   {cpr 3634

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by:  prmg  3754  difprsnss  3771  preqr1  3809  preq12b  3811  prel12  3812  pwprss  3846  pwtpss  3847  unipr  3864  intpr  3917  zfpair2  4255  elop  4276  ordtri2or2exmidlem  4575  onsucelsucexmidlem  4578  en2lp  4603  reg3exmidlemwe  4628  xpsspw  4788  acexmidlem2  5943  2oconcl  6527  exmidpw  7007  exmidpweq  7008  renfdisj  8134  fzpr  10201  maxabslemval  11552  xrmaxiflemval  11594  isprm2  12472  2lgslem4  15613  structiedg0val  15670  bj-zfpair2  15883  ss1oel2o  15965