Theorem nmotru 37411

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

Assertion

Ref	Expression
nmotru	⊢ ¬ ∃*𝑥⊤

Proof of Theorem nmotru

Step	Hyp	Ref	Expression
1		extru 2010	. . 3 ⊢ ∃𝑥⊤
2		neutru 37410	. . 3 ⊢ ¬ ∃!𝑥⊤
3		jcn 163	. . 3 ⊢ (∃𝑥⊤ → (¬ ∃!𝑥⊤ → ¬ (∃𝑥⊤ → ∃!𝑥⊤)))
4	1, 2, 3	mp2 9	. 2 ⊢ ¬ (∃𝑥⊤ → ∃!𝑥⊤)
5		moeu 2611	. 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ → ∃!𝑥⊤))
6	4, 5	mtbir 326	1 ⊢ ¬ ∃*𝑥⊤

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ⊤wtru 1571  ∃wex 1812  ∃*wmo 2565  ∃!weu 2596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-nul 5266  ax-pow 5334

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2567  df-eu 2597

This theorem is used by: (None)