Theorem sssnr 3680

Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4125. (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 3401	. . 3 |- (/) C_ { B }
2		sseq1 3120	. . 3 |- ( A = (/) -> ( A C_ { B } <-> (/) C_ { B } ) )
3	1, 2	mpbiri 167	. 2 |- ( A = (/) -> A C_ { B } )
4		eqimss 3151	. 2 |- ( A = { B } -> A C_ { B } )
5	3, 4	jaoi 705	1 |- ( ( A = (/) \/ A = { B } ) -> A C_ { B } )

Syntax hints:   -> wi 4   \/ wo 697   = wceq 1331   C_ wss 3071   (/)c0 3363   {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-in 3077  df-ss 3084  df-nul 3364

This theorem is referenced by:  pwsnss  3730