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Theorem sssn 3788
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 3478 . . . . . . 7  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
2 ssel 3187 . . . . . . . . . . 11  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  e.  { B } ) )
3 elsni 3677 . . . . . . . . . . 11  |-  ( x  e.  { B }  ->  x  =  B )
42, 3syl6 29 . . . . . . . . . 10  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  =  B ) )
5 eleq1 2356 . . . . . . . . . 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 1626 . . . . . . 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 3775 . . . . . 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 3207 . . . 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 3496 . . . 4  |-  (/)  C_  { B }
17 sseq1 3212 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  { B }  <->  (/)  C_ 
{ B } ) )
1816, 17mpbiri 224 . . 3  |-  ( A  =  (/)  ->  A  C_  { B } )
19 eqimss 3243 . . 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 1531    = wceq 1632    e. wcel 1696    C_ wss 3165   (/)c0 3468   {csn 3653
This theorem is referenced by:  eqsn  3791  snsssn  3797  pwsn  3837  frsn  4776  foconst  5478  fin1a2lem12  8053  fpwwe2lem13  8280  gsumval2  14476  0top  16737  minveclem4a  18810  ordcmp  24958  uvtx01vtx  28305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659
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