Theorem sssnr 3688

Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4133. (Contributed by Jim Kingdon, 10-Aug-2018.)

Assertion

Ref	Expression
sssnr	⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})

Proof of Theorem sssnr

Step	Hyp	Ref	Expression
1		0ss 3406	. . 3 ⊢ ∅ ⊆ {𝐵}
2		sseq1 3125	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
3	1, 2	mpbiri 167	. 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
4		eqimss 3156	. 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
5	3, 4	jaoi 706	1 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})

Syntax hints:   → wi 4   ∨ wo 698   = wceq 1332   ⊆ wss 3076  ∅c0 3368  {csn 3532

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369

This theorem is referenced by:  pwsnss  3738