Theorem neq0f 4303

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 4307 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 4302	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
3	2	notbii 323	. 2 ⊢ (¬ 𝐴 = ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
4		df-ex 1810	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
5	3, 4	bitr4i 281	1 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Ⅎwnfc 2910  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-dif 3909  df-nul 4288

This theorem is referenced by:  n0f  4304  ralfal  45862