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Theorem sssnr 3619
Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4055. (Contributed by Jim Kingdon, 10-Aug-2018.)
Assertion
Ref Expression
sssnr ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})

Proof of Theorem sssnr
StepHypRef Expression
1 0ss 3340 . . 3 ∅ ⊆ {𝐵}
2 sseq1 3062 . . 3 (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
31, 2mpbiri 167 . 2 (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
4 eqimss 3093 . 2 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
53, 4jaoi 674 1 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wo 667   = wceq 1296  wss 3013  c0 3302  {csn 3466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-nul 3303
This theorem is referenced by:  pwsnss  3669
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