Theorem ralfal 44532

Description: Two ways of expressing empty set. (Contributed by Glauco Siliprandi, 24-Jan-2024.)

Hypothesis

Ref	Expression
ralfal.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
ralfal	⊢ (𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 ⊥)

Proof of Theorem ralfal

Step	Hyp	Ref	Expression
1		df-fal 1547	. . . 4 ⊢ (⊥ ↔ ¬ ⊤)
2	1	ralbii 3090	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ∀𝑥 ∈ 𝐴 ¬ ⊤)
3		ralnex 3069	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ ⊤ ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤)
4	2, 3	bitri 275	. 2 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤)
5		rextru 3074	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)
6	5	notbii 320	. 2 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤)
7		ralfal.1	. . . 4 ⊢ Ⅎ𝑥𝐴
8	7	neq0f 4342	. . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
9	8	con1bii 356	. 2 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅)
10	4, 6, 9	3bitr2ri 300	1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1534  ⊤wtru 1535  ⊥wfal 1546  ∃wex 1774   ∈ wcel 2099  Ⅎwnfc 2879  ∀wral 3058  ∃wrex 3067  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-dif 3950  df-nul 4324

This theorem is referenced by: (None)