Theorem sssnr 3603

Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4040. (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 3325	. . 3 |- (/) C_ { B }
2		sseq1 3048	. . 3 |- ( A = (/) -> ( A C_ { B } <-> (/) C_ { B } ) )
3	1, 2	mpbiri 167	. 2 |- ( A = (/) -> A C_ { B } )
4		eqimss 3079	. 2 |- ( A = { B } -> A C_ { B } )
5	3, 4	jaoi 672	1 |- ( ( A = (/) \/ A = { B } ) -> A C_ { B } )

Syntax hints:   -> wi 4   \/ wo 665   = wceq 1290   C_ wss 3000   (/)c0 3287   {csn 3450

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-in 3006  df-ss 3013  df-nul 3288

This theorem is referenced by:  pwsnss  3653