Theorem dfpr2 3641

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 3629	. 2 |- { A , B } = ( { A } u. { B } )
2		elun 3304	. . . 4 |- ( x e. ( { A } u. { B } ) <-> ( x e. { A } \/ x e. { B } ) )
3		velsn 3639	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 3639	. . . . 5 |- ( x e. { B } <-> x = B )
5	3, 4	orbi12i 765	. . . 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 2311	. 2 |- ( { A } u. { B } ) = { x | ( x = A \/ x = B ) }
8	1, 7	eqtri 2217	1 |- { A , B } = { x | ( x = A \/ x = B ) }

Syntax hints:   \/ wo 709   = wceq 1364   e. wcel 2167   {cab 2182   u. cun 3155   {csn 3622   {cpr 3623

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629

This theorem is referenced by:  elprg  3642  nfpr  3672  pwsnss  3833  minmax  11395  xrminmax  11430