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Theorem snssiALT 28603
Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 3759. This theorem was automatically generated from snssiALTVD 28602 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snssiALT  |-  ( A  e.  B  ->  { A }  C_  B )

Proof of Theorem snssiALT
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elsn 3655 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2 eleq1a 2352 . . . 4  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
31, 2syl5bi 208 . . 3  |-  ( A  e.  B  ->  (
x  e.  { A }  ->  x  e.  B
) )
43alrimiv 1617 . 2  |-  ( A  e.  B  ->  A. x
( x  e.  { A }  ->  x  e.  B ) )
5 dfss2 3169 . 2  |-  ( { A }  C_  B  <->  A. x ( x  e. 
{ A }  ->  x  e.  B ) )
64, 5sylibr 203 1  |-  ( A  e.  B  ->  { A }  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    = wceq 1623    e. wcel 1684    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-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-sn 3646
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