Theorem xrmnfdc 10122

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 10055	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8270	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2424	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 742	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 843	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 134	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 8276	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2405	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2238	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 682	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 742	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 134	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 720	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 134	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1340	. 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 716  DECID wdc 842   \/ w3o 1004   = wceq 1398   e. wcel 2202   RRcr 8074   +oocpnf 8253   -oocmnf 8254   RR*cxr 8255

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-pnf 8258  df-mnf 8259  df-xr 8260

This theorem is referenced by:  xaddf  10123  xaddval  10124  xaddmnf1  10127  xaddcom  10140  xnegdi  10147  xpncan  10150  xleadd1a  10152  xsubge0  10160  xrmaxiflemcl  11868  xrmaxifle  11869  xrmaxiflemab  11870  xrmaxiflemlub  11871  xrmaxiflemcom  11872  xrmaxadd  11884