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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.
StepHypRef Expression
1 vex 3112 . . . . 5
21elpr 4047 . . . 4
3 elsn 4043 . . . . 5
43notbii 296 . . . 4
5 biorf 405 . . . . 5
65biimparc 487 . . . 4
72, 4, 6syl2anb 479 . . 3
8 eldif 3485 . . 3
9 elsn 4043 . . 3
107, 8, 93imtr4i 266 . 2
1110ssriv 3507 1
 Colors of variables: wff setvar class 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
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