Theorem elprg 3687

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 3686	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2951	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 3668

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 2802  df-un 3202  df-sn 3673  df-pr 3674

This theorem is referenced by:  elpr  3688  elpr2  3689  elpri  3690  eldifpr  3694  eltpg  3712  prid1g  3773  ssprss  3832  preqr1g  3847  m1expeven  10838  maxclpr  11773  minmax  11781  minclpr  11788  xrminmax  11816  perfectlem2  15714  lgslem1  15719  lgsval  15723  lgsfvalg  15724  lgsfcl2  15725  lgsval2lem  15729  lgsdir2lem4  15750  lgsdir2lem5  15751  lgsdir2  15752  lgsne0  15757  gausslemma2dlem0i  15776  2lgs  15823  2lgsoddprm  15832