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Theorem n0f 4075
Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 4078 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 2944 . 2 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 eq0f.1 . . 3 𝑥𝐴
32neq0f 4074 . 2 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
41, 3bitri 264 1 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1631  wex 1852  wcel 2145  wnfc 2900  wne 2943  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-nul 4064
This theorem is referenced by:  n0  4078  abn0  4101  cp  8918  ordtconnlem1  30310  inn0f  39763
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