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Theorem snssd 3853
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 (φA B)
Assertion
Ref Expression
snssd (φ → {A} B)

Proof of Theorem snssd
StepHypRef Expression
1 snssd.1 . 2 (φA B)
2 snssg 3844 . . 3 (A B → (A B ↔ {A} B))
31, 2syl 15 . 2 (φ → (A B ↔ {A} B))
41, 3mpbid 201 1 (φ → {A} B)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wcel 1710   wss 3257  {csn 3737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-sn 3741
This theorem is referenced by:  nchoicelem6  6294  frecsuc  6322
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