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Theorem snssiALTVD 28602
Description: Virtual deduction proof of snssiALT 28603. (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 3169 . . 3  |-  ( { A }  C_  B  <->  A. x ( x  e. 
{ A }  ->  x  e.  B ) )
2 idn1 28342 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
3 idn2 28385 . . . . . . 7  |-  (. A  e.  B ,. x  e. 
{ A }  ->.  x  e.  { A } ).
4 elsn 3655 . . . . . . 7  |-  ( x  e.  { A }  <->  x  =  A )
53, 4e2bi 28404 . . . . . 6  |-  (. A  e.  B ,. x  e. 
{ A }  ->.  x  =  A ).
6 eleq1a 2352 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
72, 5, 6e12 28499 . . . . 5  |-  (. A  e.  B ,. x  e. 
{ A }  ->.  x  e.  B ).
87in2 28377 . . . 4  |-  (. A  e.  B  ->.  ( x  e. 
{ A }  ->  x  e.  B ) ).
98gen11 28388 . . 3  |-  (. A  e.  B  ->.  A. x ( x  e.  { A }  ->  x  e.  B ) ).
10 bi2 189 . . 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 28463 . 2  |-  (. A  e.  B  ->.  { A }  C_  B ).
1211in1 28339 1  |-  ( A  e.  B  ->  { A }  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   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  df-vd1 28338  df-vd2 28347
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