Theorem ralfal 45677

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 1563	. . . 4 ⊢ (⊥ ↔ ¬ ⊤)
2	1	ralbii 3098	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ∀𝑥 ∈ 𝐴 ¬ ⊤)
3		ralnex 3078	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ ⊤ ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤)
4	2, 3	bitri 277	. 2 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤)
5		rextru 3083	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)
6	5	notbii 322	. 2 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 ⊤)
7		ralfal.1	. . . 4 ⊢ Ⅎ𝑥𝐴
8	7	neq0f 4291	. . 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 1550  ⊤wtru 1551  ⊥wfal 1562  ∃wex 1789   ∈ wcel 2132  Ⅎwnfc 2899  ∀wral 3066  ∃wrex 3076  ∅c0 4276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-11 2181  ax-12 2202  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ral 3067  df-rex 3077  df-dif 3898  df-nul 4277

This theorem is referenced by: (None)