Theorem n0f 4301

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 4305 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 2933	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		eq0f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	2	neq0f 4300	. 2 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
4	1, 3	bitri 275	1 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Ⅎwnfc 2883   ≠ wne 2932  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-dif 3904  df-nul 4286

This theorem is referenced by:  inn0f  4323  cp  9803  ordtconnlem1  34081  iinss2d  45401  r19.3rzf  45402