Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  difsn Structured version   Unicode version

Theorem difsn 3934
 Description: An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
difsn

Proof of Theorem difsn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldifsn 3928 . . 3
2 simpl 445 . . . 4
3 eleq1 2497 . . . . . . . 8
43biimpcd 217 . . . . . . 7
54necon3bd 2639 . . . . . 6
65com12 30 . . . . 5
76ancld 538 . . . 4
82, 7impbid2 197 . . 3
91, 8syl5bb 250 . 2
109eqrdv 2435 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   wceq 1653   wcel 1726   wne 2600   cdif 3318  csn 3815 This theorem is referenced by:  difsnb  3941  domdifsn  7192  domunsncan  7209  frfi  7353  infdifsn  7612  dfn2  10235  clslp  17213  nbgrassovt  21448  xrge00  24209 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-v 2959  df-dif 3324  df-sn 3821
 Copyright terms: Public domain W3C validator