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Theorem sssn 3772
Description: The subsets of a singleton. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
sssn  |-  ( A 
C_  { B }  <->  ( A  =  (/)  \/  A  =  { B } ) )

Proof of Theorem sssn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 neq0 3465 . . . . . . 7  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
2 ssel 3174 . . . . . . . . . . 11  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  e.  { B } ) )
3 elsni 3664 . . . . . . . . . . 11  |-  ( x  e.  { B }  ->  x  =  B )
42, 3syl6 29 . . . . . . . . . 10  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  =  B ) )
5 eleq1 2343 . . . . . . . . . 10  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
64, 5syl6 29 . . . . . . . . 9  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  ( x  e.  A  <->  B  e.  A ) ) )
76ibd 234 . . . . . . . 8  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  B  e.  A ) )
87exlimdv 1664 . . . . . . 7  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  B  e.  A ) )
91, 8syl5bi 208 . . . . . 6  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  B  e.  A
) )
10 snssi 3759 . . . . . 6  |-  ( B  e.  A  ->  { B }  C_  A )
119, 10syl6 29 . . . . 5  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  { B }  C_  A ) )
1211anc2li 540 . . . 4  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  ( A  C_  { B }  /\  { B }  C_  A ) ) )
13 eqss 3194 . . . 4  |-  ( A  =  { B }  <->  ( A  C_  { B }  /\  { B }  C_  A ) )
1412, 13syl6ibr 218 . . 3  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  A  =  { B } ) )
1514orrd 367 . 2  |-  ( A 
C_  { B }  ->  ( A  =  (/)  \/  A  =  { B } ) )
16 0ss 3483 . . . 4  |-  (/)  C_  { B }
17 sseq1 3199 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  { B }  <->  (/)  C_ 
{ B } ) )
1816, 17mpbiri 224 . . 3  |-  ( A  =  (/)  ->  A  C_  { B } )
19 eqimss 3230 . . 3  |-  ( A  =  { B }  ->  A  C_  { B } )
2018, 19jaoi 368 . 2  |-  ( ( A  =  (/)  \/  A  =  { B } )  ->  A  C_  { B } )
2115, 20impbii 180 1  |-  ( A 
C_  { B }  <->  ( A  =  (/)  \/  A  =  { B } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   {csn 3640
This theorem is referenced by:  eqsn  3775  snsssn  3781  pwsn  3821  frsn  4760  foconst  5462  fin1a2lem12  8037  fpwwe2lem13  8264  gsumval2  14460  0top  16721  minveclem4a  18794  ordcmp  24886  uvtx01vtx  28164
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-or 359  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-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646
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