Theorem xrnpnfmnf 39705

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

Hypotheses

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

Assertion

Ref	Expression
xrnpnfmnf	|- ( ph -> A = -oo )

Proof of Theorem xrnpnfmnf

Step	Hyp	Ref	Expression
1		xrnpnfmnf.1	. . . 4 |- ( ph -> A e. RR* )
2		xrnpnfmnf.3	. . . 4 |- ( ph -> A =/= +oo )
3	1, 2	jca 554	. . 3 |- ( ph -> ( A e. RR* /\ A =/= +oo ) )
4		xrnepnf 11952	. . 3 |- ( ( A e. RR* /\ A =/= +oo ) <-> ( A e. RR \/ A = -oo ) )
5	3, 4	sylib 208	. 2 |- ( ph -> ( A e. RR \/ A = -oo ) )
6		xrnpnfmnf.2	. 2 |- ( ph -> -. A e. RR )
7		pm2.53 388	. 2 |- ( ( A e. RR \/ A = -oo ) -> ( -. A e. RR -> A = -oo ) )
8	5, 6, 7	sylc 65	1 |- ( ph -> A = -oo )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   RRcr 9935   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-pnf 10076  df-mnf 10077  df-xr 10078

This theorem is referenced by: (None)