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

Theorem snssg 3541
Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)
Assertion
Ref Expression
snssg  |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  C_  B ) )

Proof of Theorem snssg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2145 . 2  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
2 sneq 3427 . . 3  |-  ( x  =  A  ->  { x }  =  { A } )
32sseq1d 3035 . 2  |-  ( x  =  A  ->  ( { x }  C_  B 
<->  { A }  C_  B ) )
4 vex 2613 . . 3  |-  x  e. 
_V
54snss 3534 . 2  |-  ( x  e.  B  <->  { x }  C_  B )
61, 3, 5vtoclbg 2668 1  |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    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:  snssi  3549  snssd  3550  prssg  3562  ordtri2orexmid  4294  ordtri2or2exmid  4342  relsng  4489  fvimacnvi  5333  fvimacnv  5334
  Copyright terms: Public domain W3C validator