Theorem neutru 34523

Description: There does not exist exactly one set such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Assertion

Ref	Expression
neutru	⊢ ¬ ∃!𝑥⊤

Proof of Theorem neutru

Step	Hyp	Ref	Expression
1		nexntru 34520	. 2 ⊢ ¬ ∃𝑥 ¬ ⊤
2		eunex 5308	. 2 ⊢ (∃!𝑥⊤ → ∃𝑥 ¬ ⊤)
3	1, 2	mto 196	1 ⊢ ¬ ∃!𝑥⊤

Syntax hints:  ¬ wn 3  ⊤wtru 1540  ∃wex 1783  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-nul 5225  ax-pow 5283

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-mo 2540  df-eu 2569

This theorem is referenced by:  nmotru  34524