Theorem pimltmnf2

Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -oo , is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021) (Revised by Glauco Siliprandi, 15-Dec-2024)

	Ref	Expression
Hypotheses	pimltmnf2.1	$$⊢\underset{\_}{Ⅎ}xF$$
	pimltmnf2.2	$$⊢\phi \to F:A⟶ℝ$$
Assertion	pimltmnf2	$$⊢\phi \to \left\{x\in A|F\left(x\right)<\mathrm{-\infty }\right\}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		pimltmnf2.1	$$⊢\underset{\_}{Ⅎ}xF$$
2		pimltmnf2.2	$$⊢\phi \to F:A⟶ℝ$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
4	1 3 2	pimltmnf2f	$$⊢\phi \to \left\{x\in A|F\left(x\right)<\mathrm{-\infty }\right\}=\varnothing$$