Theorem renepnf

Description: No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Assertion	renepnf	$$⊢A\in ℝ\to A\ne \mathrm{+\infty }$$

Proof

Step	Hyp	Ref	Expression
1		pnfnre	$$⊢\mathrm{+\infty }\notin ℝ$$
2	1	neli	$$⊢\neg \mathrm{+\infty }\in ℝ$$
3		eleq1	$$⊢A=\mathrm{+\infty }\to \left(A\in ℝ\leftrightarrow \mathrm{+\infty }\in ℝ\right)$$
4	2 3	mtbiri	$$⊢A=\mathrm{+\infty }\to \neg A\in ℝ$$
5	4	necon2ai	$$⊢A\in ℝ\to A\ne \mathrm{+\infty }$$