Theorem sssnr 3794

Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4246. (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 3499	. . 3 |- (/) C_ { B }
2		sseq1 3216	. . 3 |- ( A = (/) -> ( A C_ { B } <-> (/) C_ { B } ) )
3	1, 2	mpbiri 168	. 2 |- ( A = (/) -> A C_ { B } )
4		eqimss 3247	. 2 |- ( A = { B } -> A C_ { B } )
5	3, 4	jaoi 718	1 |- ( ( A = (/) \/ A = { B } ) -> A C_ { B } )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   C_ wss 3166   (/)c0 3460   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461

This theorem is referenced by:  pwsnss  3844