Theorem difsnss 3666

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 6403. (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 3220	. 2 |- ( ( A \ { B } ) u. { B } ) = ( { B } u. ( A \ { B } ) )
2		snssi 3664	. . 3 |- ( B e. A -> { B } C_ A )
3		undifss 3443	. . 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	eqsstrid 3143	1 |- ( B e. A -> ( ( A \ { B } ) u. { B } ) C_ A )

Syntax hints:   -> wi 4   e. wcel 1480   \ cdif 3068   u. cun 3069   C_ wss 3071   {csn 3527

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533

This theorem is referenced by:  fnsnsplitss  5619  dcdifsnid  6400