Theorem xrmnfdc 9779

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 9712	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 7947	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2357	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 724	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 825	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 133	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 7953	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2338	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2172	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 665	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 724	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 133	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 702	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 133	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1293	. 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 698  DECID wdc 824   \/ w3o 967   = wceq 1343   e. wcel 2136   RRcr 7752   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-pnf 7935  df-mnf 7936  df-xr 7937

This theorem is referenced by:  xaddf  9780  xaddval  9781  xaddmnf1  9784  xaddcom  9797  xnegdi  9804  xpncan  9807  xleadd1a  9809  xsubge0  9817  xrmaxiflemcl  11186  xrmaxifle  11187  xrmaxiflemab  11188  xrmaxiflemlub  11189  xrmaxiflemcom  11190  xrmaxadd  11202