Theorem elprg 3686

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 2236	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2236	. . 3 |- ( x = A -> ( x = C <-> A = C ) )
3	1, 2	orbi12d 798	. 2 |- ( x = A -> ( ( x = B \/ x = C ) <-> ( A = B \/ A = C ) ) )
4		dfpr2 3685	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2950	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  elpr  3687  elpr2  3688  elpri  3689  eldifpr  3693  eltpg  3711  prid1g  3770  ssprss  3828  preqr1g  3843  m1expeven  10803  maxclpr  11728  minmax  11736  minclpr  11743  xrminmax  11771  perfectlem2  15668  lgslem1  15673  lgsval  15677  lgsfvalg  15678  lgsfcl2  15679  lgsval2lem  15683  lgsdir2lem4  15704  lgsdir2lem5  15705  lgsdir2  15706  lgsne0  15711  gausslemma2dlem0i  15730  2lgs  15777  2lgsoddprm  15786