Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  difsnss Unicode version

Theorem difsnss 3675
 Description: If we remove a single element from a class then put it back in, we end up with a subset of the original class. If equality is decidable, we can replace subset with equality as seen in nndifsnid 6412. (Contributed by Jim Kingdon, 10-Aug-2018.)
Assertion
Ref Expression
difsnss

Proof of Theorem difsnss
StepHypRef Expression
1 uncom 3226 . 2
2 snssi 3673 . . 3
3 undifss 3449 . . 3
42, 3sylib 121 . 2
51, 4eqsstrid 3149 1
 Colors of variables: wff set class Syntax hints:   wi 4   wcel 1481   cdif 3074   cun 3075   wss 3077  csn 3533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-sn 3539 This theorem is referenced by:  fnsnsplitss  5628  dcdifsnid  6409
 Copyright terms: Public domain W3C validator