Theorem dfpr2 3692

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 3680	. 2 |- { A , B } = ( { A } u. { B } )
2		elun 3350	. . . 4 |- ( x e. ( { A } u. { B } ) <-> ( x e. { A } \/ x e. { B } ) )
3		velsn 3690	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 3690	. . . . 5 |- ( x e. { B } <-> x = B )
5	3, 4	orbi12i 772	. . . 4 |- ( ( x e. { A } \/ x e. { B } ) <-> ( x = A \/ x = B ) )
6	2, 5	bitri 184	. . 3 |- ( x e. ( { A } u. { B } ) <-> ( x = A \/ x = B ) )
7	6	abbi2i 2346	. 2 |- ( { A } u. { B } ) = { x | ( x = A \/ x = B ) }
8	1, 7	eqtri 2252	1 |- { A , B } = { x | ( x = A \/ x = B ) }

Syntax hints:   \/ wo 716   = wceq 1398   e. wcel 2202   {cab 2217   u. cun 3199   {csn 3673   {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680

This theorem is referenced by:  elprg  3693  nfpr  3723  pwsnss  3892  minmax  11853  xrminmax  11888