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Theorem ne0d 39622
 Description: If a set has elements, then it is not empty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
ne0d.1 (𝜑𝐵𝐴)
Assertion
Ref Expression
ne0d (𝜑𝐴 ≠ ∅)

Proof of Theorem ne0d
StepHypRef Expression
1 ne0d.1 . 2 (𝜑𝐵𝐴)
2 ne0i 3954 . 2 (𝐵𝐴𝐴 ≠ ∅)
31, 2syl 17 1 (𝜑𝐴 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2030   ≠ wne 2823  ∅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-v 3233  df-dif 3610  df-nul 3949 This theorem is referenced by:  uzn0d  39965  uzublem  39970  climinf2lem  40256  cnrefiisplem  40373  smfsuplem1  41338  smfsuplem3  41340
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