ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ral0 GIF version

Theorem ral0 3381
Description: Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
ral0 𝑥 ∈ ∅ 𝜑

Proof of Theorem ral0
StepHypRef Expression
1 noel 3290 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 610 . 2 (𝑥 ∈ ∅ → 𝜑)
32rgen 2428 1 𝑥 ∈ ∅ 𝜑
Colors of variables: wff set class
Syntax hints:  wcel 1438  wral 2359  c0 3286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 3001  df-nul 3287
This theorem is referenced by:  0iin  3786  po0  4136  so0  4151  we0  4186  ord0  4216  omsinds  4433  mpt0  5135  iso0  5588  ac6sfi  6604  fimax2gtri  6607  finomni  6786  uzsinds  9836  seq3f1olemp  9919  rexfiuz  10410  fimaxre2  10645  2prm  11374  bj-nntrans  11729
  Copyright terms: Public domain W3C validator