Theorem xrmnfdc 10068

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 10001	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8218	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2421	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 739	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 840	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 134	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 8224	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2402	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2236	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 679	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 739	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 134	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 717	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 134	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1337	. 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 713  DECID wdc 839   \/ w3o 1001   = wceq 1395   e. wcel 2200   RRcr 8021   +oocpnf 8201   -oocmnf 8202   RR*cxr 8203

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892  df-pnf 8206  df-mnf 8207  df-xr 8208

This theorem is referenced by:  xaddf  10069  xaddval  10070  xaddmnf1  10073  xaddcom  10086  xnegdi  10093  xpncan  10096  xleadd1a  10098  xsubge0  10106  xrmaxiflemcl  11796  xrmaxifle  11797  xrmaxiflemab  11798  xrmaxiflemlub  11799  xrmaxiflemcom  11800  xrmaxadd  11812