Theorem sssnr 3836

Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4292. (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 3533	. . 3 ⊢ ∅ ⊆ {𝐵}
2		sseq1 3250	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
3	1, 2	mpbiri 168	. 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
4		eqimss 3281	. 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
5	3, 4	jaoi 723	1 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})

Syntax hints:   → wi 4   ∨ wo 715   = wceq 1397   ⊆ wss 3200  ∅c0 3494  {csn 3669

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495

This theorem is referenced by:  pwsnss  3887