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Theorem sssnr 3515
 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 3249 . . 3 ∅ ⊆ {B}
2 sseq1 2960 . . 3 (A = ∅ → (A ⊆ {B} ↔ ∅ ⊆ {B}))
31, 2mpbiri 157 . 2 (A = ∅ → A ⊆ {B})
4 eqimss 2991 . 2 (A = {B} → A ⊆ {B})
53, 4jaoi 635 1 ((A = ∅ A = {B}) → A ⊆ {B})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 628   = wceq 1242   ⊆ wss 2911  ∅c0 3218  {csn 3367 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219 This theorem is referenced by:  pwsnss  3565
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