MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexn0 Structured version   Visualization version   GIF version

Theorem rexn0 4107
Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
Assertion
Ref Expression
rexn0 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexn0
StepHypRef Expression
1 ne0i 3954 . . 3 (𝑥𝐴𝐴 ≠ ∅)
21a1d 25 . 2 (𝑥𝐴 → (𝜑𝐴 ≠ ∅))
32rexlimiv 3056 1 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wne 2823  wrex 2942  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-nul 3949
This theorem is referenced by:  reusv2lem3  4901  eusvobj2  6683  isdrs2  16986  ismnd  17344  slwn0  18076  lbsexg  19212  iunconn  21279  grpon0  27484  filbcmb  33665  isbnd2  33712  rencldnfi  37702  iunconnlem2  39485  stoweidlem14  40549  hoidmvval0  41122  2reu4  41511
  Copyright terms: Public domain W3C validator