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Theorem snssd 3550
Description: The singleton of an element of a class is a subset of the class (deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypothesis
Ref Expression
snssd.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
snssd (𝜑 → {𝐴} ⊆ 𝐵)

Proof of Theorem snssd
StepHypRef Expression
1 snssd.1 . 2 (𝜑𝐴𝐵)
2 snssg 3541 . . 3 (𝐴𝐵 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))
31, 2syl 14 . 2 (𝜑 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))
41, 3mpbid 145 1 (𝜑 → {𝐴} ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1434  wss 2982  {csn 3416
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-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 2612  df-in 2988  df-ss 2995  df-sn 3422
This theorem is referenced by:  ecinxp  6268  xpdom3m  6399  ac6sfi  6454  undifdc  6468  en2other2  6574  un0addcl  8440  un0mulcl  8441  fseq1p1m1  9239  phicl2  10797  bj-omtrans  11018
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