Theorem dfpr2 3394

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 3382	. 2 |- { A , B } = ( { A } u. { B } )
2		elun 3084	. . . 4 |- ( x e. ( { A } u. { B } ) <-> ( x e. { A } \/ x e. { B } ) )
3		velsn 3392	. . . . 5 |- ( x e. { A } <-> x = A )
4		velsn 3392	. . . . 5 |- ( x e. { B } <-> x = B )
5	3, 4	orbi12i 681	. . . 4 |- ( ( x e. { A } \/ x e. { B } ) <-> ( x = A \/ x = B ) )
6	2, 5	bitri 173	. . 3 |- ( x e. ( { A } u. { B } ) <-> ( x = A \/ x = B ) )
7	6	abbi2i 2152	. 2 |- ( { A } u. { B } ) = { x | ( x = A \/ x = B ) }
8	1, 7	eqtri 2060	1 |- { A , B } = { x | ( x = A \/ x = B ) }

Syntax hints:   \/ wo 629   = wceq 1243   e. wcel 1393   {cab 2026   u. cun 2915   {csn 3375   {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:  elprg  3395  nfpr  3420  pwsnss  3574