Theorem n0f 3927

Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3931 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 2795	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		eq0f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	2	neq0f 3926	. 2 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
4	1, 3	bitri 264	1 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Ⅎwnfc 2751   ≠ wne 2794  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  n0  3931  abn0  3954  cp  8754  ordtconnlem1  29970  inn0f  39242