Theorem elprg 3395

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 2046	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
2		eqeq1 2046	. . 3 |- ( x = A -> ( x = C <-> A = C ) )
3	1, 2	orbi12d 707	. 2 |- ( x = A -> ( ( x = B \/ x = C ) <-> ( A = B \/ A = C ) ) )
4		dfpr2 3394	. 2 |- { B , C } = { x | ( x = B \/ x = C ) }
5	3, 4	elab2g 2689	1 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   <-> wb 98   \/ wo 629   = wceq 1243   e. wcel 1393   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by:  elpr  3396  elpr2  3397  elpri  3398  eltpg  3416  prid1g  3474  preqr1g  3537