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Theorem snssl 28369
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 3841. The proof of this theorem was automatically generated from snsslVD 28368 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 3756 . 2  |-  A  e. 
{ A }
3 ssel2 3261 . 2  |-  ( ( { A }  C_  B  /\  A  e.  { A } )  ->  A  e.  B )
42, 3mpan2 652 1  |-  ( { A }  C_  B  ->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1715   _Vcvv 2873    C_ wss 3238   {csn 3729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-in 3245  df-ss 3252  df-sn 3735
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