Theorem elprg 3614

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 2184	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2184	. . 3 |- ( x = A -> ( x = C <-> A = C ) )
3	1, 2	orbi12d 793	. 2 |- ( x = A -> ( ( x = B \/ x = C ) <-> ( A = B \/ A = C ) ) )
4		dfpr2 3613	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2886	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   {cpr 3595

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601

This theorem is referenced by:  elpr  3615  elpr2  3616  elpri  3617  eldifpr  3621  eltpg  3639  prid1g  3698  preqr1g  3768  m1expeven  10569  maxclpr  11233  minmax  11240  minclpr  11247  xrminmax  11275  lgslem1  14486  lgsval  14490  lgsfvalg  14491  lgsfcl2  14492  lgsval2lem  14496  lgsdir2lem4  14517  lgsdir2lem5  14518  lgsdir2  14519  lgsne0  14524