Theorem elprg 3689

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 2238	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2238	. . 3 |- ( x = A -> ( x = C <-> A = C ) )
3	1, 2	orbi12d 800	. 2 |- ( x = A -> ( ( x = B \/ x = C ) <-> ( A = B \/ A = C ) ) )
4		dfpr2 3688	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2953	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  elpr  3690  elpr2  3691  elpri  3692  eldifpr  3696  eltpg  3714  prid1g  3775  ssprss  3834  preqr1g  3849  m1expeven  10847  maxclpr  11782  minmax  11790  minclpr  11797  xrminmax  11825  perfectlem2  15723  lgslem1  15728  lgsval  15732  lgsfvalg  15733  lgsfcl2  15734  lgsval2lem  15738  lgsdir2lem4  15759  lgsdir2lem5  15760  lgsdir2  15761  lgsne0  15766  gausslemma2dlem0i  15785  2lgs  15832  2lgsoddprm  15841  eupth2lem1  16308