Theorem nmotru 36374

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 1975	. . 3 ⊢ ∃𝑥⊤
2		neutru 36373	. . 3 ⊢ ¬ ∃!𝑥⊤
3		jcn 162	. . 3 ⊢ (∃𝑥⊤ → (¬ ∃!𝑥⊤ → ¬ (∃𝑥⊤ → ∃!𝑥⊤)))
4	1, 2, 3	mp2 9	. 2 ⊢ ¬ (∃𝑥⊤ → ∃!𝑥⊤)
5		moeu 2586	. 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ → ∃!𝑥⊤))
6	4, 5	mtbir 323	1 ⊢ ¬ ∃*𝑥⊤

Syntax hints:  ¬ wn 3   → wi 4  ⊤wtru 1538  ∃wex 1777  ∃*wmo 2541  ∃!weu 2571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-nul 5324  ax-pow 5383

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-mo 2543  df-eu 2572

This theorem is referenced by: (None)