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

Theorem intexab 4559
Description: The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
Assertion
Ref Expression
intexab  |-  ( E. x ph  <->  |^| { x  |  ph }  e.  _V )

Proof of Theorem intexab
StepHypRef Expression
1 abn0 3754 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  E. x ph )
2 intex 4557 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  |^| { x  | 
ph }  e.  _V )
31, 2bitr3i 259 1  |-  ( E. x ph  <->  |^| { x  |  ph }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   E.wex 1671    e. wcel 1904   {cab 2457    =/= wne 2641   _Vcvv 3031   (/)c0 3722   |^|cint 4226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-int 4227
This theorem is referenced by:  intexrab  4560  tcmin  8243  cfval  8695  efgval  17445  relintabex  36258  rclexi  36293  rtrclex  36295  trclexi  36298  rtrclexi  36299
  Copyright terms: Public domain W3C validator