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Theorem rexn0 4046
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 3897 . . 3 (𝑥𝐴𝐴 ≠ ∅)
21a1d 25 . 2 (𝑥𝐴 → (𝜑𝐴 ≠ ∅))
32rexlimiv 3020 1 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wne 2790  wrex 2908  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-nul 3892
This theorem is referenced by:  reusv2lem3  4831  eusvobj2  6597  isdrs2  16860  ismnd  17218  slwn0  17951  lbsexg  19083  iunconn  21141  grpon0  27202  filbcmb  33164  isbnd2  33211  rencldnfi  36862  iunconnlem2  38651  stoweidlem14  39535  hoidmvval0  40105  2reu4  40491
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