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Theorem ne0d 39823
 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 4067 . 2 (𝐵𝐴𝐴 ≠ ∅)
31, 2syl 17 1 (𝜑𝐴 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2144   ≠ wne 2942  ∅c0 4061 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-v 3351  df-dif 3724  df-nul 4062 This theorem is referenced by:  uzn0d  40162  uzublem  40167  climinf2lem  40450  cnrefiisplem  40567  smfsuplem1  41531  smfsuplem3  41533
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