Theorem elpr 3615

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 3614	. 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 708   = wceq 1353   e. wcel 2148   _Vcvv 2739   {cpr 3595

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601

This theorem is referenced by:  prmg  3715  difprsnss  3732  preqr1  3770  preq12b  3772  prel12  3773  pwprss  3807  pwtpss  3808  unipr  3825  intpr  3878  zfpair2  4212  elop  4233  ordtri2or2exmidlem  4527  onsucelsucexmidlem  4530  en2lp  4555  reg3exmidlemwe  4580  xpsspw  4740  acexmidlem2  5875  2oconcl  6443  exmidpw  6911  exmidpweq  6912  renfdisj  8020  fzpr  10080  maxabslemval  11220  xrmaxiflemval  11261  isprm2  12120  bj-zfpair2  14850  ss1oel2o  14932