Theorem xrmnfdc 9519

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 9456	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 7738	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2303	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 706	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 803	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 133	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 7744	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2284	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2121	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 647	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 706	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 133	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 684	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 133	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1264	. 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 680  DECID wdc 802   \/ w3o 944   = wceq 1314   e. wcel 1463   RRcr 7546   +oocpnf 7721   -oocmnf 7722   RR*cxr 7723

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-uni 3703  df-pnf 7726  df-mnf 7727  df-xr 7728

This theorem is referenced by:  xaddf  9520  xaddval  9521  xaddmnf1  9524  xaddcom  9537  xnegdi  9544  xpncan  9547  xleadd1a  9549  xsubge0  9557  xrmaxiflemcl  10906  xrmaxifle  10907  xrmaxiflemab  10908  xrmaxiflemlub  10909  xrmaxiflemcom  10910  xrmaxadd  10922