Theorem xrnmnfpnf

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

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

Proof

Step	Hyp	Ref	Expression
1		xrnmnfpnf.1	|- ( ph -> A e. RR* )
2		xrnmnfpnf.2	|- ( ph -> -. A e. RR )
3		xrnmnfpnf.3	|- ( ph -> A =/= -oo )
4	1 3	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		pm2.53	|- ( ( A e. RR \/ A = +oo ) -> ( -. A e. RR -> A = +oo ) )
8	6 2 7	sylc	|- ( ph -> A = +oo )