Theorem nmotru 33777

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 1979	. . 3 ⊢ ∃𝑥⊤
2		neutru 33776	. . 3 ⊢ ¬ ∃!𝑥⊤
3		jcn 164	. . 3 ⊢ (∃𝑥⊤ → (¬ ∃!𝑥⊤ → ¬ (∃𝑥⊤ → ∃!𝑥⊤)))
4	1, 2, 3	mp2 9	. 2 ⊢ ¬ (∃𝑥⊤ → ∃!𝑥⊤)
5		moeu 2667	. 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ → ∃!𝑥⊤))
6	4, 5	mtbir 325	1 ⊢ ¬ ∃*𝑥⊤

Syntax hints:  ¬ wn 3   → wi 4  ⊤wtru 1537  ∃wex 1779  ∃*wmo 2619  ∃!weu 2652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-12 2176  ax-nul 5203  ax-pow 5259

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-mo 2621  df-eu 2653

This theorem is referenced by: (None)