Theorem xrmnfdc 9841

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 9774	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8004	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2368	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 734	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 835	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 134	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 8010	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2349	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2184	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 675	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 734	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 134	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 712	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 134	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1303	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> DECID A = -oo )
16	1, 15	sylbi 121	1 |- ( A e. RR* -> DECID A = -oo )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 708  DECID wdc 834   \/ w3o 977   = wceq 1353   e. wcel 2148   RRcr 7809   +oocpnf 7987   -oocmnf 7988   RR*cxr 7989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-pnf 7992  df-mnf 7993  df-xr 7994

This theorem is referenced by:  xaddf  9842  xaddval  9843  xaddmnf1  9846  xaddcom  9859  xnegdi  9866  xpncan  9869  xleadd1a  9871  xsubge0  9879  xrmaxiflemcl  11248  xrmaxifle  11249  xrmaxiflemab  11250  xrmaxiflemlub  11251  xrmaxiflemcom  11252  xrmaxadd  11264