Theorem difsnss 3531

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 6103. (Contributed by Jim Kingdon, 10-Aug-2018.)

Assertion

Ref	Expression
difsnss	|- ( B e. A -> ( ( A \ { B } ) u. { B } ) C_ A )

Proof of Theorem difsnss

Step	Hyp	Ref	Expression
1		uncom 3116	. 2 |- ( ( A \ { B } ) u. { B } ) = ( { B } u. ( A \ { B } ) )
2		snssi 3529	. . 3 |- ( B e. A -> { B } C_ A )
3		undifss 3323	. . 3 |- ( { B } C_ A <-> ( { B } u. ( A \ { B } ) ) C_ A )
4	2, 3	sylib 120	. 2 |- ( B e. A -> ( { B } u. ( A \ { B } ) ) C_ A )
5	1, 4	syl5eqss 3043	1 |- ( B e. A -> ( ( A \ { B } ) u. { B } ) C_ A )

Syntax hints:   -> wi 4   e. wcel 1433   \ cdif 2970   u. cun 2971   C_ wss 2973   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404

This theorem is referenced by:  nndifsnid  6103  fidifsnid  6356