Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  snssl Unicode version

Theorem snssl 28605
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 3748. The proof of this theorem was automatically generated from snsslVD 28604 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 3667 . 2  |-  A  e. 
{ A }
3 ssel2 3175 . 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 1684   _Vcvv 2788    C_ wss 3152   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-sn 3646
  Copyright terms: Public domain W3C validator