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Theorem rex0 3757
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 3746 . . 3  |-  -.  x  e.  (/)
21pm2.21i 136 . 2  |-  ( x  e.  (/)  ->  -.  ph )
32nrex 2853 1  |-  -.  E. x  e.  (/)  ph
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1897   E.wrex 2749   (/)c0 3742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ral 2753  df-rex 2754  df-v 3058  df-dif 3418  df-nul 3743
This theorem is referenced by:  0iun  4348  sup0riota  8004  cfeq0  8711  cfsuc  8712  hashge2el2difr  12670  cshws0  15120  meet0  16431  join0  16432  dya2iocuni  29153  eulerpartlemgh  29259  pmapglb2xN  33381  elpadd0  33418
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