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Theorem snssl 34049
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 4140. The proof of this theorem was automatically generated from snsslVD 34048 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 4044 . 2  |-  A  e. 
{ A }
3 ssel2 3484 . 2  |-  ( ( { A }  C_  B  /\  A  e.  { A } )  ->  A  e.  B )
42, 3mpan2 669 1  |-  ( { A }  C_  B  ->  A  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823   _Vcvv 3106    C_ wss 3461   {csn 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-in 3468  df-ss 3475  df-sn 4017
This theorem is referenced by: (None)
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