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Theorem n0f 4348
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 4352 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 2940 . 2 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 eq0f.1 . . 3 𝑥𝐴
32neq0f 4347 . 2 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
41, 3bitri 275 1 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1539  wex 1778  wcel 2107  wnfc 2889  wne 2939  c0 4332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-dif 3953  df-nul 4333
This theorem is referenced by:  inn0f  4370  cp  9932  ordtconnlem1  33924  iinss2d  45167  r19.3rzf  45168
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