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

Theorem ral0 3432
Description: Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
ral0  |-  A. x  e.  (/)  ph

Proof of Theorem ral0
StepHypRef Expression
1 noel 3335 . . 3  |-  -.  x  e.  (/)
21pm2.21i 618 . 2  |-  ( x  e.  (/)  ->  ph )
32rgen 2460 1  |-  A. x  e.  (/)  ph
Colors of variables: wff set class
Syntax hints:    e. wcel 1463   A.wral 2391   (/)c0 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-dif 3041  df-nul 3332
This theorem is referenced by:  0iin  3839  po0  4201  so0  4216  we0  4251  ord0  4281  omsinds  4503  mpt0  5218  iso0  5684  ixp0x  6586  ac6sfi  6758  fimax2gtri  6761  finomni  6978  uzsinds  10155  seq3f1olemp  10215  rexfiuz  10701  fimaxre2  10938  2prm  11704  bj-nntrans  12960
  Copyright terms: Public domain W3C validator