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Theorem snssl 31899
Description: If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss 4108. The proof of this theorem was automatically generated from snsslVD 31898 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
snssl.1  |-  A  e. 
_V
Assertion
Ref Expression
snssl  |-  ( { A }  C_  B  ->  A  e.  B )

Proof of Theorem snssl
StepHypRef Expression
1 snssl.1 . . 3  |-  A  e. 
_V
21snid 4014 . 2  |-  A  e. 
{ A }
3 ssel2 3460 . 2  |-  ( ( { A }  C_  B  /\  A  e.  { A } )  ->  A  e.  B )
42, 3mpan2 671 1  |-  ( { A }  C_  B  ->  A  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   _Vcvv 3078    C_ wss 3437   {csn 3986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-in 3444  df-ss 3451  df-sn 3987
This theorem is referenced by: (None)
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