Theorem elprg 3714

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 2241	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2241	. . 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 3713	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2967	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   {cpr 3695

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701

This theorem is referenced by:  elpr  3715  elpr2  3716  elpri  3717  eldifpr  3721  eltpg  3739  prid1g  3800  ssprss  3860  preqr1g  3875  m1expeven  10972  maxclpr  11932  minmax  11940  minclpr  11947  xrminmax  11975  perfectlem2  15994  lgslem1  15999  lgsval  16003  lgsfvalg  16004  lgsfcl2  16005  lgsval2lem  16009  lgsdir2lem4  16030  lgsdir2lem5  16031  lgsdir2  16032  lgsne0  16037  gausslemma2dlem0i  16056  2lgs  16103  2lgsoddprm  16112  eupth2lem1  16579  eupth2lem3lem4fi  16594