Theorem eq0f 4060

Description: The empty set 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 2894	. . 3 ⊢ Ⅎ𝑥∅
3	1, 2	cleqf 2920	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
4		noel 4054	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
5	4	nbn 361	. . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
6	5	albii 1888	. 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
7	3, 6	bitr4i 267	1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1622   = wceq 1624   ∈ wcel 2131  Ⅎwnfc 2881  ∅c0 4050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-v 3334  df-dif 3710  df-nul 4051

This theorem is referenced by:  neq0f  4061  eq0  4064  ab0  4086  bnj1476  31216  stoweidlem34  40746