Theorem xrpnfdc 9994

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 9928	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 8150	. . . . . 6 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2398	. . . . 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		orc 714	. . . 4 |- ( A = +oo -> ( A = +oo \/ -. A = +oo ) )
8	7, 5	sylibr 134	. . 3 |- ( A = +oo -> DECID A = +oo )
9		mnfnepnf 8158	. . . . . . 7 |- -oo =/= +oo
10	9	neii 2379	. . . . . 6 |- -. -oo = +oo
11		eqeq1 2213	. . . . . 6 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
12	10, 11	mtbiri 677	. . . . 5 |- ( A = -oo -> -. A = +oo )
13	12	olcd 736	. . . 4 |- ( A = -oo -> ( A = +oo \/ -. A = +oo ) )
14	13, 5	sylibr 134	. . 3 |- ( A = -oo -> DECID A = +oo )
15	6, 8, 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 2177   RRcr 7954   +oocpnf 8134   -oocmnf 8135   RR*cxr 8136

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-un 4493  ax-cnex 8046  ax-resscn 8047

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 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-rex 2491  df-rab 2494  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3860  df-pnf 8139  df-mnf 8140  df-xr 8141

This theorem is referenced by:  xaddf  9996  xaddval  9997  xaddpnf1  9998  xaddcom  10013  xnegdi  10020  xleadd1a  10025  xlesubadd  10035  xrmaxiflemcl  11641  xrmaxifle  11642  xrmaxiflemab  11643  xrmaxiflemlub  11644  xrmaxiflemcom  11645  xrmaxadd  11657  xblss2ps  14961  xblss2  14962