Theorem difprsnss 4165

Description: Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
difprsnss

Proof of Theorem difprsnss

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3112	. . . . 5
2	1	elpr 4047	. . . 4
3		elsn 4043	. . . . 5
4	3	notbii 296	. . . 4
5		biorf 405	. . . . 5
6	5	biimparc 487	. . . 4
7	2, 4, 6	syl2anb 479	. . 3
8		eldif 3485	. . 3
9		elsn 4043	. . 3
10	7, 8, 9	3imtr4i 266	. 2
11	10	ssriv 3507	1

Syntax hints:  -.wn 3  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  \cdif 3472  C_wss 3475  {csn 4029  {cpr 4031

This theorem is referenced by:  en2other2  8408  pmtrprfv  16478  itg11  22098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-sn 4030  df-pr 4032