Theorem xrmnfdc 9964

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 9897	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8120	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2396	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 735	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 836	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 134	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 8126	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2377	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2211	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 676	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 735	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 134	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 713	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 134	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1315	. 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 709  DECID wdc 835   \/ w3o 979   = wceq 1372   e. wcel 2175   RRcr 7923   +oocpnf 8103   -oocmnf 8104   RR*cxr 8105

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-uni 3850  df-pnf 8108  df-mnf 8109  df-xr 8110

This theorem is referenced by:  xaddf  9965  xaddval  9966  xaddmnf1  9969  xaddcom  9982  xnegdi  9989  xpncan  9992  xleadd1a  9994  xsubge0  10002  xrmaxiflemcl  11527  xrmaxifle  11528  xrmaxiflemab  11529  xrmaxiflemlub  11530  xrmaxiflemcom  11531  xrmaxadd  11543