Theorem xrpnfdc 10194

Description: An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)

Assertion

Ref	Expression
xrpnfdc	|- ( A e. RR* -> DECID A = +oo )

Proof of Theorem xrpnfdc

Step	Hyp	Ref	Expression
1		elxr 10128	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 8337	. . . . . 6 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2435	. . . . 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		orc 720	. . . 4 |- ( A = +oo -> ( A = +oo \/ -. A = +oo ) )
8	7, 5	sylibr 134	. . 3 |- ( A = +oo -> DECID A = +oo )
9		mnfnepnf 8345	. . . . . . 7 |- -oo =/= +oo
10	9	neii 2416	. . . . . 6 |- -. -oo = +oo
11		eqeq1 2241	. . . . . 6 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
12	10, 11	mtbiri 682	. . . . 5 |- ( A = -oo -> -. A = +oo )
13	12	olcd 742	. . . 4 |- ( A = -oo -> ( A = +oo \/ -. A = +oo ) )
14	13, 5	sylibr 134	. . 3 |- ( A = -oo -> DECID A = +oo )
15	6, 8, 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 2205   RRcr 8142   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-un 4559  ax-cnex 8234  ax-resscn 8235

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 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-pnf 8326  df-mnf 8327  df-xr 8328

This theorem is referenced by:  xaddf  10196  xaddval  10197  xaddpnf1  10198  xaddcom  10213  xnegdi  10220  xleadd1a  10225  xlesubadd  10235  xrmaxiflemcl  11955  xrmaxifle  11956  xrmaxiflemab  11957  xrmaxiflemlub  11958  xrmaxiflemcom  11959  xrmaxadd  11971  xblss2ps  15395  xblss2  15396