Theorem sssnr 3765

Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4214. (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 3473	. . 3 ⊢ ∅ ⊆ {𝐵}
2		sseq1 3190	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
3	1, 2	mpbiri 168	. 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
4		eqimss 3221	. 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
5	3, 4	jaoi 717	1 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})

Syntax hints:   → wi 4   ∨ wo 709   = wceq 1363   ⊆ wss 3141  ∅c0 3434  {csn 3604

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154  df-nul 3435

This theorem is referenced by:  pwsnss  3815