Theorem elprg 3709

Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)

Assertion

Ref	Expression
elprg	|- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Proof of Theorem elprg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2239	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2239	. . 3 |- ( x = A -> ( x = C <-> A = C ) )
3	1, 2	orbi12d 801	. 2 |- ( x = A -> ( ( x = B \/ x = C ) <-> ( A = B \/ A = C ) ) )
4		dfpr2 3708	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2964	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2203   {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:  elpr  3710  elpr2  3711  elpri  3712  eldifpr  3716  eltpg  3734  prid1g  3795  ssprss  3855  preqr1g  3870  m1expeven  10948  maxclpr  11907  minmax  11915  minclpr  11922  xrminmax  11950  perfectlem2  15868  lgslem1  15873  lgsval  15877  lgsfvalg  15878  lgsfcl2  15879  lgsval2lem  15883  lgsdir2lem4  15904  lgsdir2lem5  15905  lgsdir2  15906  lgsne0  15911  gausslemma2dlem0i  15930  2lgs  15977  2lgsoddprm  15986  eupth2lem1  16453  eupth2lem3lem4fi  16468