Theorem empty 1939

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 1583	. . 3 ⊢ (⊥ ↔ ¬ ⊤)
2	1	albii 1852	. 2 ⊢ (∀𝑥⊥ ↔ ∀𝑥 ¬ ⊤)
3		alnex 1814	. 2 ⊢ (∀𝑥 ¬ ⊤ ↔ ¬ ∃𝑥⊤)
4	2, 3	bitr2i 279	1 ⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1568  ⊤wtru 1571  ⊥wfal 1582  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-fal 1583  df-ex 1813

This theorem is used by:  bj-cbveaw  37375