ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sssnr GIF version

Theorem sssnr 3571
Description: Empty set and the singleton itself are subsets of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
Assertion
Ref Expression
sssnr ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})

Proof of Theorem sssnr
StepHypRef Expression
1 0ss 3303 . . 3 ∅ ⊆ {𝐵}
2 sseq1 3031 . . 3 (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
31, 2mpbiri 166 . 2 (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
4 eqimss 3062 . 2 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
53, 4jaoi 669 1 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662   = wceq 1285  wss 2984  c0 3269  {csn 3422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-in 2990  df-ss 2997  df-nul 3270
This theorem is referenced by:  pwsnss  3621
  Copyright terms: Public domain W3C validator