Theorem elprg 3511

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 2119	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2119	. . 3 |- ( x = A -> ( x = C <-> A = C ) )
3	1, 2	orbi12d 765	. 2 |- ( x = A -> ( ( x = B \/ x = C ) <-> ( A = B \/ A = C ) ) )
4		dfpr2 3510	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2798	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 680   = wceq 1312   e. wcel 1461   {cpr 3492

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-sn 3497  df-pr 3498

This theorem is referenced by:  elpr  3512  elpr2  3513  elpri  3514  eltpg  3533  prid1g  3591  preqr1g  3657  m1expeven  10227  maxclpr  10880  minmax  10887  minclpr  10894  xrminmax  10920