Theorem elprg 3547

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 2146	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2146	. . 3 |- ( x = A -> ( x = C <-> A = C ) )
3	1, 2	orbi12d 782	. 2 |- ( x = A -> ( ( x = B \/ x = C ) <-> ( A = B \/ A = C ) ) )
4		dfpr2 3546	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2831	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   {cpr 3528

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534

This theorem is referenced by:  elpr  3548  elpr2  3549  elpri  3550  eltpg  3569  prid1g  3627  preqr1g  3693  m1expeven  10340  maxclpr  10994  minmax  11001  minclpr  11008  xrminmax  11034