Theorem dfpr2 3551

Description: Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)

Assertion

Ref	Expression
dfpr2	|- { A , B } = { x | ( x = A \/ x = B ) }

Distinct variable groups:   x, A   x, B

Proof of Theorem dfpr2

Step	Hyp	Ref	Expression
1		df-pr 3539	. 2 |- { A , B } = ( { A } u. { B } )
2		elun 3222	. . . 4 |- ( x e. ( { A } u. { B } ) <-> ( x e. { A } \/ x e. { B } ) )
3		velsn 3549	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 3549	. . . . 5 |- ( x e. { B } <-> x = B )
5	3, 4	orbi12i 754	. . . 4 |- ( ( x e. { A } \/ x e. { B } ) <-> ( x = A \/ x = B ) )
6	2, 5	bitri 183	. . 3 |- ( x e. ( { A } u. { B } ) <-> ( x = A \/ x = B ) )
7	6	abbi2i 2255	. 2 |- ( { A } u. { B } ) = { x | ( x = A \/ x = B ) }
8	1, 7	eqtri 2161	1 |- { A , B } = { x | ( x = A \/ x = B ) }

Syntax hints:   \/ wo 698   = wceq 1332   e. wcel 1481   {cab 2126   u. cun 3074   {csn 3532   {cpr 3533

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539

This theorem is referenced by:  elprg  3552  nfpr  3581  pwsnss  3738  minmax  11033  xrminmax  11066