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Theorem snssd 3500
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 3491 . . 3 (A B → (A B ↔ {A} ⊆ B))
31, 2syl 14 . 2 (φ → (A B ↔ {A} ⊆ B))
41, 3mpbid 135 1 (φ → {A} ⊆ B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1390  wss 2911  {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-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-in 2918  df-ss 2925  df-sn 3373
This theorem is referenced by:  ecinxp  6117  xpdom3m  6244  un0addcl  7991  un0mulcl  7992  fseq1p1m1  8726  bj-omtrans  9416
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