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Theorem snssiALTVD 33728
Description: Virtual deduction proof of snssiALT 33729. (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 3488 . . 3  |-  ( { A }  C_  B  <->  A. x ( x  e. 
{ A }  ->  x  e.  B ) )
2 idn1 33452 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
3 idn2 33500 . . . . . . 7  |-  (. A  e.  B ,. x  e. 
{ A }  ->.  x  e.  { A } ).
4 elsn 4046 . . . . . . 7  |-  ( x  e.  { A }  <->  x  =  A )
53, 4e2bi 33519 . . . . . 6  |-  (. A  e.  B ,. x  e. 
{ A }  ->.  x  =  A ).
6 eleq1a 2540 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
72, 5, 6e12 33622 . . . . 5  |-  (. A  e.  B ,. x  e. 
{ A }  ->.  x  e.  B ).
87in2 33492 . . . 4  |-  (. A  e.  B  ->.  ( x  e. 
{ A }  ->  x  e.  B ) ).
98gen11 33503 . . 3  |-  (. A  e.  B  ->.  A. x ( x  e.  { A }  ->  x  e.  B ) ).
10 bi2 198 . . 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 33578 . 2  |-  (. A  e.  B  ->.  { A }  C_  B ).
1211in1 33449 1  |-  ( A  e.  B  ->  { A }  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1393    = wceq 1395    e. wcel 1819    C_ wss 3471   {csn 4032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-in 3478  df-ss 3485  df-sn 4033  df-vd1 33448  df-vd2 33456
This theorem is referenced by: (None)
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