Theorem nltpnft 9627

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 9593	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 7837	. . . . 5 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2330	. . . 4 |- ( A e. RR -> -. A = +oo )
4		ltpnf 9597	. . . . 5 |- ( A e. RR -> A < +oo )
5		notnot 619	. . . . 5 |- ( A < +oo -> -. -. A < +oo )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. A < +oo )
7	3, 6	2falsed 692	. . 3 |- ( A e. RR -> ( A = +oo <-> -. A < +oo ) )
8		id 19	. . . 4 |- ( A = +oo -> A = +oo )
9		pnfxr 7842	. . . . . 6 |- +oo e. RR*
10		xrltnr 9596	. . . . . 6 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . 5 |- -. +oo < +oo
12		breq1 3940	. . . . 5 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 665	. . . 4 |- ( A = +oo -> -. A < +oo )
14	8, 13	2thd 174	. . 3 |- ( A = +oo -> ( A = +oo <-> -. A < +oo ) )
15		mnfnepnf 7845	. . . . . 6 |- -oo =/= +oo
16	15	neii 2311	. . . . 5 |- -. -oo = +oo
17		eqeq1 2147	. . . . 5 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
18	16, 17	mtbiri 665	. . . 4 |- ( A = -oo -> -. A = +oo )
19		mnfltpnf 9601	. . . . . . 7 |- -oo < +oo
20		breq1 3940	. . . . . . 7 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
21	19, 20	mpbiri 167	. . . . . 6 |- ( A = -oo -> A < +oo )
22	21	necon3bi 2359	. . . . 5 |- ( -. A < +oo -> A =/= -oo )
23	22	necon2bi 2364	. . . 4 |- ( A = -oo -> -. -. A < +oo )
24	18, 23	2falsed 692	. . 3 |- ( A = -oo -> ( A = +oo <-> -. A < +oo ) )
25	7, 14, 24	3jaoi 1282	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A = +oo <-> -. A < +oo ) )
26	1, 25	sylbi 120	1 |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ w3o 962   = wceq 1332   e. wcel 1481   class class class wbr 3937   RRcr 7643   +oocpnf 7821   -oocmnf 7822   RR*cxr 7823   < clt 7824

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-pre-ltirr 7756

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829

This theorem is referenced by:  npnflt  9628  xgepnf  9629  xrmaxiflemlub  11049