MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexn0 Structured version   Unicode version

Theorem rexn0 3722
Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
Assertion
Ref Expression
rexn0  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rexn0
StepHypRef Expression
1 ne0i 3626 . . 3  |-  ( x  e.  A  ->  A  =/=  (/) )
21a1d 23 . 2  |-  ( x  e.  A  ->  ( ph  ->  A  =/=  (/) ) )
32rexlimiv 2816 1  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    =/= wne 2598   E.wrex 2698   (/)c0 3620
This theorem is referenced by:  reusv2lem3  4718  reusv7OLD  4727  eusvobj2  6574  isdrs2  14388  ismnd  14684  slwn0  15241  lbsexg  16228  iuncon  17483  grpon0  21782  subofld  24237  filbcmb  26433  isbnd2  26483  rencldnfi  26873  stoweidlem14  27730  2reu4  27935  iunconlem2  28984
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-nul 3621
  Copyright terms: Public domain W3C validator