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Theorem sssn 4130
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 3749 . . . . . . 7  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
2 ssel 3436 . . . . . . . . . . 11  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  e.  { B } ) )
3 elsni 3997 . . . . . . . . . . 11  |-  ( x  e.  { B }  ->  x  =  B )
42, 3syl6 31 . . . . . . . . . 10  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  =  B ) )
5 eleq1 2474 . . . . . . . . . 10  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
64, 5syl6 31 . . . . . . . . 9  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  ( x  e.  A  <->  B  e.  A ) ) )
76ibd 243 . . . . . . . 8  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  B  e.  A ) )
87exlimdv 1745 . . . . . . 7  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  B  e.  A ) )
91, 8syl5bi 217 . . . . . 6  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  B  e.  A
) )
10 snssi 4116 . . . . . 6  |-  ( B  e.  A  ->  { B }  C_  A )
119, 10syl6 31 . . . . 5  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  { B }  C_  A ) )
1211anc2li 555 . . . 4  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  ( A  C_  { B }  /\  { B }  C_  A ) ) )
13 eqss 3457 . . . 4  |-  ( A  =  { B }  <->  ( A  C_  { B }  /\  { B }  C_  A ) )
1412, 13syl6ibr 227 . . 3  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  A  =  { B } ) )
1514orrd 376 . 2  |-  ( A 
C_  { B }  ->  ( A  =  (/)  \/  A  =  { B } ) )
16 0ss 3768 . . . 4  |-  (/)  C_  { B }
17 sseq1 3463 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  { B }  <->  (/)  C_ 
{ B } ) )
1816, 17mpbiri 233 . . 3  |-  ( A  =  (/)  ->  A  C_  { B } )
19 eqimss 3494 . . 3  |-  ( A  =  { B }  ->  A  C_  { B } )
2018, 19jaoi 377 . 2  |-  ( ( A  =  (/)  \/  A  =  { B } )  ->  A  C_  { B } )
2115, 20impbii 187 1  |-  ( A 
C_  { B }  <->  ( A  =  (/)  \/  A  =  { B } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842    C_ wss 3414   (/)c0 3738   {csn 3972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-v 3061  df-dif 3417  df-in 3421  df-ss 3428  df-nul 3739  df-sn 3973
This theorem is referenced by:  eqsn  4133  snsssn  4140  pwsn  4185  frsn  4894  foconst  5789  fin1a2lem12  8823  fpwwe2lem13  9050  gsumval2  16231  0top  19777  minveclem4a  22137  uvtx01vtx  24909  locfinref  28297  ordcmp  30679
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