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Theorem rexn0 4267
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 4121 . . 3 (𝑥𝐴𝐴 ≠ ∅)
21a1d 25 . 2 (𝑥𝐴 → (𝜑𝐴 ≠ ∅))
32rexlimiv 3208 1 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  wne 2971  wrex 3090  c0 4115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-nul 4116
This theorem is referenced by:  reusv2lem3  5070  eusvobj2  6871  isdrs2  17254  ismnd  17612  slwn0  18343  lbsexg  19487  iunconn  21560  grpon0  27882  filbcmb  34023  isbnd2  34069  rencldnfi  38171  iunconnlem2  39931  stoweidlem14  40974  hoidmvval0  41547  2reu4  41967
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