Theorem xrpnfdc 9917

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 9851	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 8074	. . . . . 6 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2388	. . . . 5 |- ( A e. RR -> -. A = +oo )
4	3	olcd 735	. . . 4 |- ( A e. RR -> ( A = +oo \/ -. A = +oo ) )
5		df-dc 836	. . . 4 |- (DECID A = +oo <-> ( A = +oo \/ -. A = +oo ) )
6	4, 5	sylibr 134	. . 3 |- ( A e. RR -> DECID A = +oo )
7		orc 713	. . . 4 |- ( A = +oo -> ( A = +oo \/ -. A = +oo ) )
8	7, 5	sylibr 134	. . 3 |- ( A = +oo -> DECID A = +oo )
9		mnfnepnf 8082	. . . . . . 7 |- -oo =/= +oo
10	9	neii 2369	. . . . . 6 |- -. -oo = +oo
11		eqeq1 2203	. . . . . 6 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
12	10, 11	mtbiri 676	. . . . 5 |- ( A = -oo -> -. A = +oo )
13	12	olcd 735	. . . 4 |- ( A = -oo -> ( A = +oo \/ -. A = +oo ) )
14	13, 5	sylibr 134	. . 3 |- ( A = -oo -> DECID A = +oo )
15	6, 8, 14	3jaoi 1314	. 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 709  DECID wdc 835   \/ w3o 979   = wceq 1364   e. wcel 2167   RRcr 7878   +oocpnf 8058   -oocmnf 8059   RR*cxr 8060

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-un 4468  ax-cnex 7970  ax-resscn 7971

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-pnf 8063  df-mnf 8064  df-xr 8065

This theorem is referenced by:  xaddf  9919  xaddval  9920  xaddpnf1  9921  xaddcom  9936  xnegdi  9943  xleadd1a  9948  xlesubadd  9958  xrmaxiflemcl  11410  xrmaxifle  11411  xrmaxiflemab  11412  xrmaxiflemlub  11413  xrmaxiflemcom  11414  xrmaxadd  11426  xblss2ps  14640  xblss2  14641