Theorem xrmnfdc 10077

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 10010	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8227	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2423	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 741	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 842	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 134	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 8233	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2404	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2238	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 681	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 741	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 134	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 719	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 134	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1339	. 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 715  DECID wdc 841   \/ w3o 1003   = wceq 1397   e. wcel 2202   RRcr 8030   +oocpnf 8210   -oocmnf 8211   RR*cxr 8212

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-pnf 8215  df-mnf 8216  df-xr 8217

This theorem is referenced by:  xaddf  10078  xaddval  10079  xaddmnf1  10082  xaddcom  10095  xnegdi  10102  xpncan  10105  xleadd1a  10107  xsubge0  10115  xrmaxiflemcl  11805  xrmaxifle  11806  xrmaxiflemab  11807  xrmaxiflemlub  11808  xrmaxiflemcom  11809  xrmaxadd  11821