Theorem nltpnft 8884

Description: An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)

Assertion

Ref	Expression
nltpnft	|- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )

Proof of Theorem nltpnft

Step	Hyp	Ref	Expression
1		elxr 8850	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 7166	. . . . 5 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2266	. . . 4 |- ( A e. RR -> -. A = +oo )
4		ltpnf 8856	. . . . 5 |- ( A e. RR -> A < +oo )
5		notnot 591	. . . . 5 |- ( A < +oo -> -. -. A < +oo )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. A < +oo )
7	3, 6	2falsed 650	. . 3 |- ( A e. RR -> ( A = +oo <-> -. A < +oo ) )
8		id 19	. . . 4 |- ( A = +oo -> A = +oo )
9		pnfxr 8846	. . . . . 6 |- +oo e. RR*
10		xrltnr 8855	. . . . . 6 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 7	. . . . 5 |- -. +oo < +oo
12		breq1 3788	. . . . 5 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 632	. . . 4 |- ( A = +oo -> -. A < +oo )
14	8, 13	2thd 173	. . 3 |- ( A = +oo -> ( A = +oo <-> -. A < +oo ) )
15		mnfnepnf 8852	. . . . . 6 |- -oo =/= +oo
16	15	neii 2247	. . . . 5 |- -. -oo = +oo
17		eqeq1 2087	. . . . 5 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
18	16, 17	mtbiri 632	. . . 4 |- ( A = -oo -> -. A = +oo )
19		mnfltpnf 8860	. . . . . . 7 |- -oo < +oo
20		breq1 3788	. . . . . . 7 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
21	19, 20	mpbiri 166	. . . . . 6 |- ( A = -oo -> A < +oo )
22	21	necon3bi 2295	. . . . 5 |- ( -. A < +oo -> A =/= -oo )
23	22	necon2bi 2300	. . . 4 |- ( A = -oo -> -. -. A < +oo )
24	18, 23	2falsed 650	. . 3 |- ( A = -oo -> ( A = +oo <-> -. A < +oo ) )
25	7, 14, 24	3jaoi 1234	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A = +oo <-> -. A < +oo ) )
26	1, 25	sylbi 119	1 |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ w3o 918   = wceq 1284   e. wcel 1433   class class class wbr 3785   RRcr 6980   +oocpnf 7150   -oocmnf 7151   RR*cxr 7152   < clt 7153

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-pre-ltirr 7088

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158

This theorem is referenced by: (None)