Theorem xrmnfdc 9965

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 9898	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8121	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2397	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 736	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 837	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 134	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 8127	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2378	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2212	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 677	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 736	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 134	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 714	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 134	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1316	. 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 710  DECID wdc 836   \/ w3o 980   = wceq 1373   e. wcel 2176   RRcr 7924   +oocpnf 8104   -oocmnf 8105   RR*cxr 8106

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-pnf 8109  df-mnf 8110  df-xr 8111

This theorem is referenced by:  xaddf  9966  xaddval  9967  xaddmnf1  9970  xaddcom  9983  xnegdi  9990  xpncan  9993  xleadd1a  9995  xsubge0  10003  xrmaxiflemcl  11556  xrmaxifle  11557  xrmaxiflemab  11558  xrmaxiflemlub  11559  xrmaxiflemcom  11560  xrmaxadd  11572