Theorem elpr 3710

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 3709	. 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 716   = wceq 1398   e. wcel 2203   _Vcvv 2813   {cpr 3690

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696

This theorem is referenced by:  prmg  3814  difprsnss  3832  preqr1  3872  preq12b  3874  prel12  3875  pwprss  3910  pwtpss  3911  unipr  3928  intpr  3981  zfpair2  4323  elop  4347  ordtri2or2exmidlem  4648  onsucelsucexmidlem  4651  en2lp  4676  reg3exmidlemwe  4701  xpsspw  4862  acexmidlem2  6047  2oconcl  6672  exmidpw  7168  exmidpweq  7169  renfdisj  8333  fzpr  10411  maxabslemval  11893  xrmaxiflemval  11935  isprm2  12814  2lgslem4  15976  structiedg0val  16035  bj-zfpair2  16680  ss1oel2o  16761