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Theorem snssiALTVD 37223
Description: Virtual deduction proof of snssiALT 37224. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snssiALTVD  |-  ( A  e.  B  ->  { A }  C_  B )

Proof of Theorem snssiALTVD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3421 . . 3  |-  ( { A }  C_  B  <->  A. x ( x  e. 
{ A }  ->  x  e.  B ) )
2 idn1 36944 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
3 idn2 36992 . . . . . . 7  |-  (. A  e.  B ,. x  e. 
{ A }  ->.  x  e.  { A } ).
4 elsn 3982 . . . . . . 7  |-  ( x  e.  { A }  <->  x  =  A )
53, 4e2bi 37011 . . . . . 6  |-  (. A  e.  B ,. x  e. 
{ A }  ->.  x  =  A ).
6 eleq1a 2524 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
72, 5, 6e12 37111 . . . . 5  |-  (. A  e.  B ,. x  e. 
{ A }  ->.  x  e.  B ).
87in2 36984 . . . 4  |-  (. A  e.  B  ->.  ( x  e. 
{ A }  ->  x  e.  B ) ).
98gen11 36995 . . 3  |-  (. A  e.  B  ->.  A. x ( x  e.  { A }  ->  x  e.  B ) ).
10 biimpr 202 . . 3  |-  ( ( { A }  C_  B 
<-> 
A. x ( x  e.  { A }  ->  x  e.  B ) )  ->  ( A. x ( x  e. 
{ A }  ->  x  e.  B )  ->  { A }  C_  B
) )
111, 9, 10e01 37070 . 2  |-  (. A  e.  B  ->.  { A }  C_  B ).
1211in1 36941 1  |-  ( A  e.  B  ->  { A }  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1442    = wceq 1444    e. wcel 1887    C_ wss 3404   {csn 3968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-in 3411  df-ss 3418  df-sn 3969  df-vd1 36940  df-vd2 36948
This theorem is referenced by: (None)
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