Theorem eq0f 4255

Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by BJ, 15-Jul-2021.)

Hypothesis

Ref	Expression
eq0f.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
eq0f	⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)

Proof of Theorem eq0f

Step	Hyp	Ref	Expression
1		eq0f.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfcv 2955	. . 3 ⊢ Ⅎ𝑥∅
3	1, 2	cleqf 2983	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
4		noel 4247	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
5	4	nbn 376	. . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
6	5	albii 1821	. 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
7	3, 6	bitr4i 281	1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  ∅c0 4243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-dif 3884  df-nul 4244

This theorem is referenced by:  neq0f  4256  eq0OLD  4261  ab0  4287  bnj1476  32229  stoweidlem34  42676