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

Theorem rabsnt 3658
Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
rabsnt.1  |-  B  e. 
_V
rabsnt.2  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rabsnt  |-  ( { x  e.  A  |  ph }  =  { B }  ->  ps )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem rabsnt
StepHypRef Expression
1 rabsnt.1 . . . 4  |-  B  e. 
_V
21snid 3614 . . 3  |-  B  e. 
{ B }
3 id 19 . . 3  |-  ( { x  e.  A  |  ph }  =  { B }  ->  { x  e.  A  |  ph }  =  { B } )
42, 3eleqtrrid 2260 . 2  |-  ( { x  e.  A  |  ph }  =  { B }  ->  B  e.  {
x  e.  A  |  ph } )
5 rabsnt.2 . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
65elrab 2886 . . 3  |-  ( B  e.  { x  e.  A  |  ph }  <->  ( B  e.  A  /\  ps ) )
76simprbi 273 . 2  |-  ( B  e.  { x  e.  A  |  ph }  ->  ps )
84, 7syl 14 1  |-  ( { x  e.  A  |  ph }  =  { B }  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348    e. wcel 2141   {crab 2452   _Vcvv 2730   {csn 3583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-sn 3589
This theorem is referenced by:  ontr2exmid  4509  onsucsssucexmid  4511  ordsoexmid  4546  unfiexmid  6895
  Copyright terms: Public domain W3C validator