Theorem xrpnfdc 9769

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 9703	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 7937	. . . . . 6 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2355	. . . . 5 |- ( A e. RR -> -. A = +oo )
4	3	olcd 724	. . . 4 |- ( A e. RR -> ( A = +oo \/ -. A = +oo ) )
5		df-dc 825	. . . 4 |- (DECID A = +oo <-> ( A = +oo \/ -. A = +oo ) )
6	4, 5	sylibr 133	. . 3 |- ( A e. RR -> DECID A = +oo )
7		orc 702	. . . 4 |- ( A = +oo -> ( A = +oo \/ -. A = +oo ) )
8	7, 5	sylibr 133	. . 3 |- ( A = +oo -> DECID A = +oo )
9		mnfnepnf 7945	. . . . . . 7 |- -oo =/= +oo
10	9	neii 2336	. . . . . 6 |- -. -oo = +oo
11		eqeq1 2171	. . . . . 6 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
12	10, 11	mtbiri 665	. . . . 5 |- ( A = -oo -> -. A = +oo )
13	12	olcd 724	. . . 4 |- ( A = -oo -> ( A = +oo \/ -. A = +oo ) )
14	13, 5	sylibr 133	. . 3 |- ( A = -oo -> DECID A = +oo )
15	6, 8, 14	3jaoi 1292	. 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 698  DECID wdc 824   \/ w3o 966   = wceq 1342   e. wcel 2135   RRcr 7743   +oocpnf 7921   -oocmnf 7922   RR*cxr 7923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-un 4405  ax-cnex 7835  ax-resscn 7836

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-rex 2448  df-rab 2451  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-uni 3784  df-pnf 7926  df-mnf 7927  df-xr 7928

This theorem is referenced by:  xaddf  9771  xaddval  9772  xaddpnf1  9773  xaddcom  9788  xnegdi  9795  xleadd1a  9800  xlesubadd  9810  xrmaxiflemcl  11172  xrmaxifle  11173  xrmaxiflemab  11174  xrmaxiflemlub  11175  xrmaxiflemcom  11176  xrmaxadd  11188  xblss2ps  12945  xblss2  12946