Theorem neq0f 4294

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 4298 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 4293	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
3	2	notbii 323	. 2 ⊢ (¬ 𝐴 = ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
4		df-ex 1812	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
5	3, 4	bitr4i 281	1 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Ⅎwnfc 2907  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-dif 4047  df-nul 4279

This theorem is used by:  n0f  4295  ralfal  46270