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

Theorem rex0 3920
 Description: Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
rex0 ¬ ∃𝑥 ∈ ∅ 𝜑

Proof of Theorem rex0
StepHypRef Expression
1 noel 3901 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 116 . 2 (𝑥 ∈ ∅ → ¬ 𝜑)
32nrex 2996 1 ¬ ∃𝑥 ∈ ∅ 𝜑
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 1987  ∃wrex 2909  ∅c0 3897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-dif 3563  df-nul 3898 This theorem is referenced by:  0iun  4550  sup0riota  8331  cfeq0  9038  cfsuc  9039  hashge2el2difr  13217  cshws0  15751  meet0  17077  join0  17078  dya2iocuni  30168  eulerpartlemgh  30263  pmapglb2xN  34577  elpadd0  34614  sprsymrelfvlem  41058
 Copyright terms: Public domain W3C validator