Theorem elprg 3658

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 2213	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2213	. . 3 |- ( x = A -> ( x = C <-> A = C ) )
3	1, 2	orbi12d 795	. 2 |- ( x = A -> ( ( x = B \/ x = C ) <-> ( A = B \/ A = C ) ) )
4		dfpr2 3657	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2924	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2177   {cpr 3639

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645

This theorem is referenced by:  elpr  3659  elpr2  3660  elpri  3661  eldifpr  3665  eltpg  3683  prid1g  3742  preqr1g  3813  m1expeven  10753  maxclpr  11608  minmax  11616  minclpr  11623  xrminmax  11651  perfectlem2  15547  lgslem1  15552  lgsval  15556  lgsfvalg  15557  lgsfcl2  15558  lgsval2lem  15562  lgsdir2lem4  15583  lgsdir2lem5  15584  lgsdir2  15585  lgsne0  15590  gausslemma2dlem0i  15609  2lgs  15656  2lgsoddprm  15665