Theorem xrmnfdc 9626

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 9563	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 7814	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2329	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 723	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 820	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 133	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 7820	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2310	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2146	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 664	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 723	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 133	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 701	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 133	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1281	. 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 697  DECID wdc 819   \/ w3o 961   = wceq 1331   e. wcel 1480   RRcr 7619   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-pnf 7802  df-mnf 7803  df-xr 7804

This theorem is referenced by:  xaddf  9627  xaddval  9628  xaddmnf1  9631  xaddcom  9644  xnegdi  9651  xpncan  9654  xleadd1a  9656  xsubge0  9664  xrmaxiflemcl  11014  xrmaxifle  11015  xrmaxiflemab  11016  xrmaxiflemlub  11017  xrmaxiflemcom  11018  xrmaxadd  11030