Theorem mon 1929

Description: There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)

Assertion

Ref	Expression
mon	⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem mon

Step	Hyp	Ref	Expression
1		ax-in2 545	. 2 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
2		df-mo 1904	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
3	1, 2	sylibr 137	1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1381  ∃!weu 1900  ∃*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-mo 1904

This theorem is referenced by:  moexexdc  1984