MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sssn Structured version   Unicode version

Theorem sssn 3950
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 3631 . . . . . . 7  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
2 ssel 3335 . . . . . . . . . . 11  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  e.  { B } ) )
3 elsni 3831 . . . . . . . . . . 11  |-  ( x  e.  { B }  ->  x  =  B )
42, 3syl6 31 . . . . . . . . . 10  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  =  B ) )
5 eleq1 2496 . . . . . . . . . 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 235 . . . . . . . 8  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  B  e.  A ) )
87exlimdv 1646 . . . . . . 7  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  B  e.  A ) )
91, 8syl5bi 209 . . . . . 6  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  B  e.  A
) )
10 snssi 3935 . . . . . 6  |-  ( B  e.  A  ->  { B }  C_  A )
119, 10syl6 31 . . . . 5  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  { B }  C_  A ) )
1211anc2li 541 . . . 4  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  ( A  C_  { B }  /\  { B }  C_  A ) ) )
13 eqss 3356 . . . 4  |-  ( A  =  { B }  <->  ( A  C_  { B }  /\  { B }  C_  A ) )
1412, 13syl6ibr 219 . . 3  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  A  =  { B } ) )
1514orrd 368 . 2  |-  ( A 
C_  { B }  ->  ( A  =  (/)  \/  A  =  { B } ) )
16 0ss 3649 . . . 4  |-  (/)  C_  { B }
17 sseq1 3362 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  { B }  <->  (/)  C_ 
{ B } ) )
1816, 17mpbiri 225 . . 3  |-  ( A  =  (/)  ->  A  C_  { B } )
19 eqimss 3393 . . 3  |-  ( A  =  { B }  ->  A  C_  { B } )
2018, 19jaoi 369 . 2  |-  ( ( A  =  (/)  \/  A  =  { B } )  ->  A  C_  { B } )
2115, 20impbii 181 1  |-  ( A 
C_  { B }  <->  ( A  =  (/)  \/  A  =  { B } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    C_ wss 3313   (/)c0 3621   {csn 3807
This theorem is referenced by:  eqsn  3953  snsssn  3960  pwsn  4002  frsn  4941  foconst  5657  fin1a2lem12  8284  fpwwe2lem13  8510  gsumval2  14776  0top  17041  minveclem4a  19324  uvtx01vtx  21494  ordcmp  26190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2951  df-dif 3316  df-in 3320  df-ss 3327  df-nul 3622  df-sn 3813
  Copyright terms: Public domain W3C validator