Theorem empty 1906

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 1549	. . 3 ⊢ (⊥ ↔ ¬ ⊤)
2	1	albii 1819	. 2 ⊢ (∀𝑥⊥ ↔ ∀𝑥 ¬ ⊤)
3		alnex 1781	. 2 ⊢ (∀𝑥 ¬ ⊤ ↔ ¬ ∃𝑥⊤)
4	2, 3	bitr2i 278	1 ⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥)

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1534  ⊤wtru 1537  ⊥wfal 1548  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 209  df-fal 1549  df-ex 1780

This theorem is referenced by: (None)