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Theorem sssn 4185
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 3795 . . . . . . 7  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
2 ssel 3498 . . . . . . . . . . 11  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  e.  { B } ) )
3 elsni 4052 . . . . . . . . . . 11  |-  ( x  e.  { B }  ->  x  =  B )
42, 3syl6 33 . . . . . . . . . 10  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  =  B ) )
5 eleq1 2539 . . . . . . . . . 10  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
64, 5syl6 33 . . . . . . . . 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 1700 . . . . . . 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 4171 . . . . . 6  |-  ( B  e.  A  ->  { B }  C_  A )
119, 10syl6 33 . . . . 5  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  { B }  C_  A ) )
1211anc2li 557 . . . 4  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  ( A  C_  { B }  /\  { B }  C_  A ) ) )
13 eqss 3519 . . . 4  |-  ( A  =  { B }  <->  ( A  C_  { B }  /\  { B }  C_  A ) )
1412, 13syl6ibr 227 . . 3  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  A  =  { B } ) )
1514orrd 378 . 2  |-  ( A 
C_  { B }  ->  ( A  =  (/)  \/  A  =  { B } ) )
16 0ss 3814 . . . 4  |-  (/)  C_  { B }
17 sseq1 3525 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  { B }  <->  (/)  C_ 
{ B } ) )
1816, 17mpbiri 233 . . 3  |-  ( A  =  (/)  ->  A  C_  { B } )
19 eqimss 3556 . . 3  |-  ( A  =  { B }  ->  A  C_  { B } )
2018, 19jaoi 379 . 2  |-  ( ( A  =  (/)  \/  A  =  { B } )  ->  A  C_  { B } )
2115, 20impbii 188 1  |-  ( A 
C_  { B }  <->  ( A  =  (/)  \/  A  =  { B } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    C_ wss 3476   (/)c0 3785   {csn 4027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028
This theorem is referenced by:  eqsn  4188  snsssn  4195  pwsn  4239  frsn  5069  foconst  5804  fin1a2lem12  8787  fpwwe2lem13  9016  gsumval2  15826  0top  19251  minveclem4a  21580  uvtx01vtx  24168  ordcmp  29489
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