Theorem n0f 4299

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 4303 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	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
n0f	⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

Proof of Theorem n0f

Step	Hyp	Ref	Expression
1		df-ne 2960	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		eq0f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	2	neq0f 4298	. 2 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
4	1, 3	bitri 278	1 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Ⅎwnfc 2911   ≠ wne 2959  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-dif 4050  df-nul 4282

This theorem is used by:  inn0f  4322  cp  9991  ordtconnlem1  34667  iinss2d  46445  r19.3rzf  46446