Metamath Proof Explorer


Theorem nmotru

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

Ref Expression
Assertion nmotru ¬ ∃* 𝑥

Proof

Step Hyp Ref Expression
1 extru 𝑥
2 neutru ¬ ∃! 𝑥
3 jcn ( ∃ 𝑥 ⊤ → ( ¬ ∃! 𝑥 ⊤ → ¬ ( ∃ 𝑥 ⊤ → ∃! 𝑥 ⊤ ) ) )
4 1 2 3 mp2 ¬ ( ∃ 𝑥 ⊤ → ∃! 𝑥 ⊤ )
5 moeu ( ∃* 𝑥 ⊤ ↔ ( ∃ 𝑥 ⊤ → ∃! 𝑥 ⊤ ) )
6 4 5 mtbir ¬ ∃* 𝑥