Theorem xrnpnfmnf

Description: An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022)

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

Proof

Step	Hyp	Ref	Expression
1		xrnpnfmnf.1	|- ( ph -> A e. RR* )
2		xrnpnfmnf.2	|- ( ph -> -. A e. RR )
3		xrnpnfmnf.3	|- ( ph -> A =/= +oo )
4	1 3	jca	|- ( ph -> ( A e. RR* /\ A =/= +oo ) )
5		xrnepnf	|- ( ( 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 )