Theorem sssnr 3831

Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4286. (Contributed by Jim Kingdon, 10-Aug-2018.)

Assertion

Ref	Expression
sssnr	|- ( ( A = (/) \/ A = { B } ) -> A C_ { B } )

Proof of Theorem sssnr

Step	Hyp	Ref	Expression
1		0ss 3530	. . 3 |- (/) C_ { B }
2		sseq1 3247	. . 3 |- ( A = (/) -> ( A C_ { B } <-> (/) C_ { B } ) )
3	1, 2	mpbiri 168	. 2 |- ( A = (/) -> A C_ { B } )
4		eqimss 3278	. 2 |- ( A = { B } -> A C_ { B } )
5	3, 4	jaoi 721	1 |- ( ( A = (/) \/ A = { B } ) -> A C_ { B } )

Syntax hints:   -> wi 4   \/ wo 713   = wceq 1395   C_ wss 3197   (/)c0 3491   {csn 3666

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492

This theorem is referenced by:  pwsnss  3882