Theorem sssnr 3800

Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4254. (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 3503	. . 3 ⊢ ∅ ⊆ {𝐵}
2		sseq1 3220	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
3	1, 2	mpbiri 168	. 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
4		eqimss 3251	. 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
5	3, 4	jaoi 718	1 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})

Syntax hints:   → wi 4   ∨ wo 710   = wceq 1373   ⊆ wss 3170  ∅c0 3464  {csn 3638

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-in 3176  df-ss 3183  df-nul 3465

This theorem is referenced by:  pwsnss  3850