Theorem xrmnfdc 9800

Description: An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)

Assertion

Ref	Expression
xrmnfdc	|- ( A e. RR* -> DECID A = -oo )

Proof of Theorem xrmnfdc

Step	Hyp	Ref	Expression
1		elxr 9733	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 7968	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2361	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 729	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 830	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 133	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 7974	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2342	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2177	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 670	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 729	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 133	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 707	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 133	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1298	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> DECID A = -oo )
16	1, 15	sylbi 120	1 |- ( A e. RR* -> DECID A = -oo )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 703  DECID wdc 829   \/ w3o 972   = wceq 1348   e. wcel 2141   RRcr 7773   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-pnf 7956  df-mnf 7957  df-xr 7958

This theorem is referenced by:  xaddf  9801  xaddval  9802  xaddmnf1  9805  xaddcom  9818  xnegdi  9825  xpncan  9828  xleadd1a  9830  xsubge0  9838  xrmaxiflemcl  11208  xrmaxifle  11209  xrmaxiflemab  11210  xrmaxiflemlub  11211  xrmaxiflemcom  11212  xrmaxadd  11224