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Theorem ral0 3387
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 3291 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 611 . 2 (𝑥 ∈ ∅ → 𝜑)
32rgen 2429 1 𝑥 ∈ ∅ 𝜑
Colors of variables: wff set class
Syntax hints:  wcel 1439  wral 2360  c0 3287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2622  df-dif 3002  df-nul 3288
This theorem is referenced by:  0iin  3794  po0  4147  so0  4162  we0  4197  ord0  4227  omsinds  4448  mpt0  5154  iso0  5610  ixp0x  6497  ac6sfi  6668  fimax2gtri  6671  finomni  6857  uzsinds  9909  seq3f1olemp  9992  rexfiuz  10483  fimaxre2  10719  2prm  11448  bj-nntrans  12119
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