Theorem nltpnft 8960

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 8928	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 7228	. . . . 5 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2267	. . . 4 |- ( A e. RR -> -. A = +oo )
4		ltpnf 8932	. . . . 5 |- ( A e. RR -> A < +oo )
5		notnot 592	. . . . 5 |- ( A < +oo -> -. -. A < +oo )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. A < +oo )
7	3, 6	2falsed 651	. . 3 |- ( A e. RR -> ( A = +oo <-> -. A < +oo ) )
8		id 19	. . . 4 |- ( A = +oo -> A = +oo )
9		pnfxr 7233	. . . . . 6 |- +oo e. RR*
10		xrltnr 8931	. . . . . 6 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 7	. . . . 5 |- -. +oo < +oo
12		breq1 3796	. . . . 5 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 633	. . . 4 |- ( A = +oo -> -. A < +oo )
14	8, 13	2thd 173	. . 3 |- ( A = +oo -> ( A = +oo <-> -. A < +oo ) )
15		mnfnepnf 7236	. . . . . 6 |- -oo =/= +oo
16	15	neii 2248	. . . . 5 |- -. -oo = +oo
17		eqeq1 2088	. . . . 5 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
18	16, 17	mtbiri 633	. . . 4 |- ( A = -oo -> -. A = +oo )
19		mnfltpnf 8936	. . . . . . 7 |- -oo < +oo
20		breq1 3796	. . . . . . 7 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
21	19, 20	mpbiri 166	. . . . . 6 |- ( A = -oo -> A < +oo )
22	21	necon3bi 2296	. . . . 5 |- ( -. A < +oo -> A =/= -oo )
23	22	necon2bi 2301	. . . 4 |- ( A = -oo -> -. -. A < +oo )
24	18, 23	2falsed 651	. . 3 |- ( A = -oo -> ( A = +oo <-> -. A < +oo ) )
25	7, 14, 24	3jaoi 1235	. 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 919   = wceq 1285   e. wcel 1434   class class class wbr 3793   RRcr 7042   +oocpnf 7212   -oocmnf 7213   RR*cxr 7214   < clt 7215

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-pre-ltirr 7150

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220

This theorem is referenced by: (None)