Theorem difsnss 3764

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 6560. (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 3303	. 2 |- ( ( A \ { B } ) u. { B } ) = ( { B } u. ( A \ { B } ) )
2		snssi 3762	. . 3 |- ( B e. A -> { B } C_ A )
3		undifss 3527	. . 3 |- ( { B } C_ A <-> ( { B } u. ( A \ { B } ) ) C_ A )
4	2, 3	sylib 122	. 2 |- ( B e. A -> ( { B } u. ( A \ { B } ) ) C_ A )
5	1, 4	eqsstrid 3225	1 |- ( B e. A -> ( ( A \ { B } ) u. { B } ) C_ A )

Syntax hints:   -> wi 4   e. wcel 2164   \ cdif 3150   u. cun 3151   C_ wss 3153   {csn 3618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624

This theorem is referenced by:  fnsnsplitss  5757  dcdifsnid  6557