Theorem elprg 3693

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 2238	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2238	. . 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 3692	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2954	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   {cpr 3674

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680

This theorem is referenced by:  elpr  3694  elpr2  3695  elpri  3696  eldifpr  3700  eltpg  3718  prid1g  3779  ssprss  3839  preqr1g  3854  m1expeven  10894  maxclpr  11845  minmax  11853  minclpr  11860  xrminmax  11888  perfectlem2  15797  lgslem1  15802  lgsval  15806  lgsfvalg  15807  lgsfcl2  15808  lgsval2lem  15812  lgsdir2lem4  15833  lgsdir2lem5  15834  lgsdir2  15835  lgsne0  15840  gausslemma2dlem0i  15859  2lgs  15906  2lgsoddprm  15915  eupth2lem1  16382  eupth2lem3lem4fi  16397