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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
StepHypRef Expression
1 0ss 3473 . . 3 ∅ ⊆ {𝐵}
2 sseq1 3190 . . 3 (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
31, 2mpbiri 168 . 2 (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
4 eqimss 3221 . 2 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
53, 4jaoi 717 1 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
Colors of variables: wff set class
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
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