Theorem neq0f 4341

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 4345 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 4340	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
3	2	notbii 320	. 2 ⊢ (¬ 𝐴 = ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
4		df-ex 1781	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
5	3, 4	bitr4i 278	1 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1538   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Ⅎwnfc 2882  ∅c0 4322

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-dif 3951  df-nul 4323

This theorem is referenced by:  n0f  4342  neq0OLD  4348  ralfal  44316