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Theorem snssb 3726
Description: Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
Assertion
Ref Expression
snssb  |-  ( { A }  C_  B  <->  ( A  e.  _V  ->  A  e.  B ) )

Proof of Theorem snssb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3145 . 2  |-  ( { A }  C_  B  <->  A. x ( x  e. 
{ A }  ->  x  e.  B ) )
2 velsn 3610 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
32imbi1i 238 . . 3  |-  ( ( x  e.  { A }  ->  x  e.  B
)  <->  ( x  =  A  ->  x  e.  B ) )
43albii 1470 . 2  |-  ( A. x ( x  e. 
{ A }  ->  x  e.  B )  <->  A. x
( x  =  A  ->  x  e.  B
) )
5 eleq1 2240 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
65pm5.74i 180 . . . 4  |-  ( ( x  =  A  ->  x  e.  B )  <->  ( x  =  A  ->  A  e.  B )
)
76albii 1470 . . 3  |-  ( A. x ( x  =  A  ->  x  e.  B )  <->  A. x
( x  =  A  ->  A  e.  B
) )
8 19.23v 1883 . . 3  |-  ( A. x ( x  =  A  ->  A  e.  B )  <->  ( E. x  x  =  A  ->  A  e.  B ) )
9 isset 2744 . . . . 5  |-  ( A  e.  _V  <->  E. x  x  =  A )
109bicomi 132 . . . 4  |-  ( E. x  x  =  A  <-> 
A  e.  _V )
1110imbi1i 238 . . 3  |-  ( ( E. x  x  =  A  ->  A  e.  B )  <->  ( A  e.  _V  ->  A  e.  B ) )
127, 8, 113bitri 206 . 2  |-  ( A. x ( x  =  A  ->  x  e.  B )  <->  ( A  e.  _V  ->  A  e.  B ) )
131, 4, 123bitri 206 1  |-  ( { A }  C_  B  <->  ( A  e.  _V  ->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351    = wceq 1353   E.wex 1492    e. wcel 2148   _Vcvv 2738    C_ wss 3130   {csn 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-sn 3599
This theorem is referenced by:  snssg  3727
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