Theorem sssnr 3841

Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4298. (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 3535	. . 3 ⊢ ∅ ⊆ {𝐵}
2		sseq1 3251	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
3	1, 2	mpbiri 168	. 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
4		eqimss 3282	. 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
5	3, 4	jaoi 724	1 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})

Syntax hints:   → wi 4   ∨ wo 716   = wceq 1398   ⊆ wss 3201  ∅c0 3496  {csn 3673

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497

This theorem is referenced by:  pwsnss  3892