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Theorem snssl 28655
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 3890. The proof of this theorem was automatically generated from snsslVD 28654 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 3805 . 2  |-  A  e. 
{ A }
3 ssel2 3307 . 2  |-  ( ( { A }  C_  B  /\  A  e.  { A } )  ->  A  e.  B )
42, 3mpan2 653 1  |-  ( { A }  C_  B  ->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   _Vcvv 2920    C_ wss 3284   {csn 3778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-in 3291  df-ss 3298  df-sn 3784
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