Theorem elprg 3580

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 2164	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2164	. . 3 |- ( x = A -> ( x = C <-> A = C ) )
3	1, 2	orbi12d 783	. 2 |- ( x = A -> ( ( x = B \/ x = C ) <-> ( A = B \/ A = C ) ) )
4		dfpr2 3579	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2859	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 698   = wceq 1335   e. wcel 2128   {cpr 3561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567

This theorem is referenced by:  elpr  3581  elpr2  3582  elpri  3583  eldifpr  3587  eltpg  3604  prid1g  3663  preqr1g  3729  m1expeven  10448  maxclpr  11104  minmax  11111  minclpr  11118  xrminmax  11144