MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rex0 Structured version   Unicode version

Theorem rex0 3794
Description: Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
rex0  |-  -.  E. x  e.  (/)  ph

Proof of Theorem rex0
StepHypRef Expression
1 noel 3784 . . 3  |-  -.  x  e.  (/)
21pm2.21i 131 . 2  |-  ( x  e.  (/)  ->  -.  ph )
32nrex 2914 1  |-  -.  E. x  e.  (/)  ph
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1762   E.wrex 2810   (/)c0 3780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-v 3110  df-dif 3474  df-nul 3781
This theorem is referenced by:  0iun  4377  cfeq0  8627  cfsuc  8628  cshws0  14435  meet0  15615  join0  15616  dya2iocuni  27882  eulerpartlemgh  27945  pmapglb2xN  34445  elpadd0  34482
  Copyright terms: Public domain W3C validator