Theorem elpr 3643

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 3642	. 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 709   = wceq 1364   e. wcel 2167   _Vcvv 2763   {cpr 3623

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629

This theorem is referenced by:  prmg  3743  difprsnss  3760  preqr1  3798  preq12b  3800  prel12  3801  pwprss  3835  pwtpss  3836  unipr  3853  intpr  3906  zfpair2  4243  elop  4264  ordtri2or2exmidlem  4562  onsucelsucexmidlem  4565  en2lp  4590  reg3exmidlemwe  4615  xpsspw  4775  acexmidlem2  5919  2oconcl  6497  exmidpw  6969  exmidpweq  6970  renfdisj  8086  fzpr  10152  maxabslemval  11373  xrmaxiflemval  11415  isprm2  12285  2lgslem4  15344  bj-zfpair2  15556  ss1oel2o  15638