Theorem xrred

Description: An extended real that is neither minus infinity, nor plus infinity, is real. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	xrred.1	|- ( ph -> A e. RR* )
	xrred.2	|- ( ph -> A =/= -oo )
	xrred.3	|- ( ph -> A =/= +oo )
Assertion	xrred	|- ( ph -> A e. RR )

Proof

Step	Hyp	Ref	Expression
1		xrred.1	|- ( ph -> A e. RR* )
2		xrred.2	|- ( ph -> A =/= -oo )
3		xrred.3	|- ( ph -> A =/= +oo )
4	1 2	jca	|- ( ph -> ( A e. RR* /\ A =/= -oo ) )
5		xrnemnf	|- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
6	4 5	sylib	|- ( ph -> ( A e. RR \/ A = +oo ) )
7	3	neneqd	|- ( ph -> -. A = +oo )
8		pm2.53	|- ( ( A e. RR \/ A = +oo ) -> ( -. A e. RR -> A = +oo ) )
9	8	con1d	|- ( ( A e. RR \/ A = +oo ) -> ( -. A = +oo -> A e. RR ) )
10	6 7 9	sylc	|- ( ph -> A e. RR )