Theorem mont 32408

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

Assertion

Ref	Expression
mont	⊢ ¬ ∃*𝑥⊤

Proof of Theorem mont

Step	Hyp	Ref	Expression
1		extt 32403	. . 3 ⊢ ∃𝑥⊤
2		unnt 32407	. . 3 ⊢ ¬ ∃!𝑥⊤
3		mth8 158	. . 3 ⊢ (∃𝑥⊤ → (¬ ∃!𝑥⊤ → ¬ (∃𝑥⊤ → ∃!𝑥⊤)))
4	1, 2, 3	mp2 9	. 2 ⊢ ¬ (∃𝑥⊤ → ∃!𝑥⊤)
5		df-mo 2475	. 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ → ∃!𝑥⊤))
6	4, 5	mtbir 313	1 ⊢ ¬ ∃*𝑥⊤

Syntax hints:  ¬ wn 3   → wi 4  ⊤wtru 1484  ∃wex 1704  ∃!weu 2470  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)