Theorem sssnr 3779

Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4231. (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 3485	. . 3 |- (/) C_ { B }
2		sseq1 3202	. . 3 |- ( A = (/) -> ( A C_ { B } <-> (/) C_ { B } ) )
3	1, 2	mpbiri 168	. 2 |- ( A = (/) -> A C_ { B } )
4		eqimss 3233	. 2 |- ( A = { B } -> A C_ { B } )
5	3, 4	jaoi 717	1 |- ( ( A = (/) \/ A = { B } ) -> A C_ { B } )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   C_ wss 3153   (/)c0 3446   {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-in 3159  df-ss 3166  df-nul 3447

This theorem is referenced by:  pwsnss  3829