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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 ⊤