Theorem rzal 3931

Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
rzal

Distinct variable group:   ,

Proof of Theorem rzal

Step	Hyp	Ref	Expression
1		ne0i 3790	. . . 4
2	1	necon2bi 2694	. . 3
3	2	pm2.21d 106	. 2
4	3	ralrimiv 2869	1

Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  A.wral 2807  c0 3784

This theorem is referenced by:  ralidm  3933  rgenz  3935  ralf0  3936  raaan  3937  raaanv  3938  iinrab2  4393  riinrab  4406  reusv2lem2  4654  cnvpo  5550  dffi3  7911  brdom3  8927  dedekind  9765  fimaxre2  10516  efgs1  16753  opnnei  19621  axcontlem12  24278  ubthlem1  25786  nofulllem2  29463  mbfresfi  30061  bddiblnc  30085  blbnd  30283  rrnequiv  30331  upbdrech2  31508  stoweidlem9  31791  fourierdlem31  31920  raaan2  32180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3478  df-nul 3785