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Theorem snssiALT 28919
Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 3775. This theorem was automatically generated from snssiALTVD 28918 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 3668 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2 eleq1a 2365 . . . 4  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
31, 2syl5bi 208 . . 3  |-  ( A  e.  B  ->  (
x  e.  { A }  ->  x  e.  B
) )
43alrimiv 1621 . 2  |-  ( A  e.  B  ->  A. x
( x  e.  { A }  ->  x  e.  B ) )
5 dfss2 3182 . 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 1530    = wceq 1632    e. wcel 1696    C_ wss 3165   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179  df-sn 3659
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