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Theorem n0f 4343
Description: A class is nonempty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 4347 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by NM, 17-Oct-2003.)
Hypothesis
Ref Expression
eq0f.1 𝑥𝐴
Assertion
Ref Expression
n0f (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)

Proof of Theorem n0f
StepHypRef Expression
1 df-ne 2942 . 2 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 eq0f.1 . . 3 𝑥𝐴
32neq0f 4342 . 2 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
41, 3bitri 275 1 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1542  wex 1782  wcel 2107  wnfc 2884  wne 2941  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-dif 3952  df-nul 4324
This theorem is referenced by:  n0OLD  4350  abn0OLD  4382  cp  9886  ordtconnlem1  32904  inn0f  43760  iinss2d  43851  r19.3rzf  43852
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