Theorem empty 1909

Description: Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.)

Assertion

Ref	Expression
empty	⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥)

Proof of Theorem empty

Step	Hyp	Ref	Expression
1		df-fal 1552	. . 3 ⊢ (⊥ ↔ ¬ ⊤)
2	1	albii 1822	. 2 ⊢ (∀𝑥⊥ ↔ ∀𝑥 ¬ ⊤)
3		alnex 1784	. 2 ⊢ (∀𝑥 ¬ ⊤ ↔ ¬ ∃𝑥⊤)
4	2, 3	bitr2i 275	1 ⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1537  ⊤wtru 1540  ⊥wfal 1551  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-fal 1552  df-ex 1783

This theorem is referenced by: (None)