Theorem ralfal 45739

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 1573	. . . 4 ⊢ (⊥ ↔ ¬ ⊤)
2	1	ralbii 3108	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ∀𝑥 ∈ 𝐴 ¬ ⊤)
3		ralnex 3088	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ ⊤ ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤)
4	2, 3	bitri 277	. 2 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤)
5		rextru 3093	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)
6	5	notbii 322	. 2 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤)
7		ralfal.1	. . . 4 ⊢ Ⅎ𝑥𝐴
8	7	neq0f 4300	. . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
9	8	con1bii 358	. 2 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅)
10	4, 6, 9	3bitr2ri 302	1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1560  ⊤wtru 1561  ⊥wfal 1572  ∃wex 1799   ∈ wcel 2142  Ⅎwnfc 2909  ∀wral 3076  ∃wrex 3086  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-dif 3907  df-nul 4286

This theorem is referenced by: (None)