Theorem n0f 4342

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 4346 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 2941	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		eq0f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	2	neq0f 4341	. 2 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
4	1, 3	bitri 274	1 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Ⅎwnfc 2883   ≠ wne 2940  ∅c0 4322

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-dif 3951  df-nul 4323

This theorem is referenced by:  n0OLD  4349  abn0OLD  4381  cp  9885  ordtconnlem1  32899  inn0f  43750  iinss2d  43841  r19.3rzf  43842