ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sssnm Unicode version

Theorem sssnm 3566
Description: The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
Assertion
Ref Expression
sssnm  |-  ( E. x  x  e.  A  ->  ( A  C_  { B } 
<->  A  =  { B } ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem sssnm
StepHypRef Expression
1 ssel 3002 . . . . . . . . . 10  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  e.  { B } ) )
2 elsni 3434 . . . . . . . . . 10  |-  ( x  e.  { B }  ->  x  =  B )
31, 2syl6 33 . . . . . . . . 9  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  =  B ) )
4 eleq1 2145 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
53, 4syl6 33 . . . . . . . 8  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  ( x  e.  A  <->  B  e.  A ) ) )
65ibd 176 . . . . . . 7  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  B  e.  A ) )
76exlimdv 1742 . . . . . 6  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  B  e.  A ) )
8 snssi 3549 . . . . . 6  |-  ( B  e.  A  ->  { B }  C_  A )
97, 8syl6 33 . . . . 5  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  { B }  C_  A ) )
109anc2li 322 . . . 4  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  ( A  C_  { B }  /\  { B }  C_  A ) ) )
11 eqss 3023 . . . 4  |-  ( A  =  { B }  <->  ( A  C_  { B }  /\  { B }  C_  A ) )
1210, 11syl6ibr 160 . . 3  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  A  =  { B } ) )
1312com12 30 . 2  |-  ( E. x  x  e.  A  ->  ( A  C_  { B }  ->  A  =  { B } ) )
14 eqimss 3060 . 2  |-  ( A  =  { B }  ->  A  C_  { B } )
1513, 14impbid1 140 1  |-  ( E. x  x  e.  A  ->  ( A  C_  { B } 
<->  A  =  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434    C_ wss 2982   {csn 3416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2988  df-ss 2995  df-sn 3422
This theorem is referenced by:  eqsnm  3567  exmid01  3988
  Copyright terms: Public domain W3C validator