Theorem xrpnfdc 9512

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 9450	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 7731	. . . . . 6 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2301	. . . . 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		orc 684	. . . 4 |- ( A = +oo -> ( A = +oo \/ -. A = +oo ) )
8	7, 5	sylibr 133	. . 3 |- ( A = +oo -> DECID A = +oo )
9		mnfnepnf 7739	. . . . . . 7 |- -oo =/= +oo
10	9	neii 2282	. . . . . 6 |- -. -oo = +oo
11		eqeq1 2119	. . . . . 6 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
12	10, 11	mtbiri 647	. . . . 5 |- ( A = -oo -> -. A = +oo )
13	12	olcd 706	. . . 4 |- ( A = -oo -> ( A = +oo \/ -. A = +oo ) )
14	13, 5	sylibr 133	. . 3 |- ( A = -oo -> DECID A = +oo )
15	6, 8, 14	3jaoi 1262	. 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 942   = wceq 1312   e. wcel 1461   RRcr 7540   +oocpnf 7715   -oocmnf 7716   RR*cxr 7717

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 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-un 4313  ax-cnex 7630  ax-resscn 7631

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-rex 2394  df-rab 2397  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-uni 3701  df-pnf 7720  df-mnf 7721  df-xr 7722

This theorem is referenced by:  xaddf  9514  xaddval  9515  xaddpnf1  9516  xaddcom  9531  xnegdi  9538  xleadd1a  9543  xlesubadd  9553  xrmaxiflemcl  10900  xrmaxifle  10901  xrmaxiflemab  10902  xrmaxiflemlub  10903  xrmaxiflemcom  10904  xrmaxadd  10916  xblss2ps  12387  xblss2  12388