Theorem dfpr2 3595

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 3583	. 2 |- { A , B } = ( { A } u. { B } )
2		elun 3263	. . . 4 |- ( x e. ( { A } u. { B } ) <-> ( x e. { A } \/ x e. { B } ) )
3		velsn 3593	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 3593	. . . . 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 2281	. 2 |- ( { A } u. { B } ) = { x | ( x = A \/ x = B ) }
8	1, 7	eqtri 2186	1 |- { A , B } = { x | ( x = A \/ x = B ) }

Syntax hints:   \/ wo 698   = wceq 1343   e. wcel 2136   {cab 2151   u. cun 3114   {csn 3576   {cpr 3577

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  elprg  3596  nfpr  3626  pwsnss  3783  minmax  11171  xrminmax  11206