Theorem sssnr 3515

Description: Empty set and the singleton itself are subsets of a singleton. (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 3249	. . 3 |- (/) C_ {B }
2		sseq1 2960	. . 3 |- (A = (/) -> (A C_ {B } <-> (/) C_ {B }))
3	1, 2	mpbiri 157	. 2 |- (A = (/) -> A C_ {B })
4		eqimss 2991	. 2 |- (A = {B } -> A C_ {B })
5	3, 4	jaoi 635	1 |- ((A = (/) \/ A = {B }) -> A C_ {B })

Syntax hints:   -> wi 4   \/ wo 628   = wceq 1242   C_ wss 2911   (/)c0 3218   {csn 3367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219

This theorem is referenced by:  pwsnss  3565