Theorem difsnss 3613

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 6333. (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 3167	. 2 |- ( ( A \ { B } ) u. { B } ) = ( { B } u. ( A \ { B } ) )
2		snssi 3611	. . 3 |- ( B e. A -> { B } C_ A )
3		undifss 3390	. . 3 |- ( { B } C_ A <-> ( { B } u. ( A \ { B } ) ) C_ A )
4	2, 3	sylib 121	. 2 |- ( B e. A -> ( { B } u. ( A \ { B } ) ) C_ A )
5	1, 4	syl5eqss 3093	1 |- ( B e. A -> ( ( A \ { B } ) u. { B } ) C_ A )

Syntax hints:   -> wi 4   e. wcel 1448   \ cdif 3018   u. cun 3019   C_ wss 3021   {csn 3474

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480

This theorem is referenced by:  fnsnsplitss  5551  dcdifsnid  6330