Theorem neq0f 4309

Description: A class is not empty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of neq0 4312 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by BJ, 15-Jul-2021.)

Hypothesis

Ref	Expression
eq0f.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
neq0f	⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

Proof of Theorem neq0f

Step	Hyp	Ref	Expression
1		eq0f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	eq0f 4308	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
3	2	notbii 322	. 2 ⊢ (¬ 𝐴 = ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
4		df-ex 1780	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
5	3, 4	bitr4i 280	1 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1534   = wceq 1536  ∃wex 1779   ∈ wcel 2113  Ⅎwnfc 2964  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-dif 3942  df-nul 4295

This theorem is referenced by:  n0f  4310  neq0  4312