Theorem elpr 3726

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 3725	. 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 720   = wceq 1402   e. wcel 2209   _Vcvv 2821   {cpr 3706

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712

This theorem is referenced by:  prmg  3830  difprsnss  3848  preqr1  3888  preq12b  3890  prel12  3891  pwprss  3926  pwtpss  3927  unipr  3944  intpr  3997  zfpair2  4342  elop  4366  ordtri2or2exmidlem  4668  onsucelsucexmidlem  4671  en2lp  4696  reg3exmidlemwe  4721  xpsspw  4882  acexmidlem2  6072  2oconcl  6702  exmidpw  7205  exmidpweq  7206  renfdisj  8375  fzpr  10462  maxabslemval  11952  xrmaxiflemval  11994  isprm2  12873  2lgslem4  16136  structiedg0val  16195  bj-zfpair2  16850  ss1oel2o  16931