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Theorem n0f 4243
 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 4247 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 2952 . 2 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 eq0f.1 . . 3 𝑥𝐴
32neq0f 4242 . 2 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
41, 3bitri 278 1 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Ⅎwnfc 2899   ≠ wne 2951  ∅c0 4227 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-dif 3863  df-nul 4228 This theorem is referenced by:  n0OLD  4250  abn0OLD  4281  cp  9358  ordtconnlem1  31399  inn0f  42108
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