Theorem nltpnft 10147

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 10109	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 8321	. . . . 5 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2433	. . . 4 |- ( A e. RR -> -. A = +oo )
4		ltpnf 10113	. . . . 5 |- ( A e. RR -> A < +oo )
5		notnot 634	. . . . 5 |- ( A < +oo -> -. -. A < +oo )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. A < +oo )
7	3, 6	2falsed 710	. . 3 |- ( A e. RR -> ( A = +oo <-> -. A < +oo ) )
8		id 19	. . . 4 |- ( A = +oo -> A = +oo )
9		pnfxr 8326	. . . . . 6 |- +oo e. RR*
10		xrltnr 10112	. . . . . 6 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . 5 |- -. +oo < +oo
12		breq1 4112	. . . . 5 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 682	. . . 4 |- ( A = +oo -> -. A < +oo )
14	8, 13	2thd 175	. . 3 |- ( A = +oo -> ( A = +oo <-> -. A < +oo ) )
15		mnfnepnf 8329	. . . . . 6 |- -oo =/= +oo
16	15	neii 2414	. . . . 5 |- -. -oo = +oo
17		eqeq1 2239	. . . . 5 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
18	16, 17	mtbiri 682	. . . 4 |- ( A = -oo -> -. A = +oo )
19		mnfltpnf 10118	. . . . . . 7 |- -oo < +oo
20		breq1 4112	. . . . . . 7 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
21	19, 20	mpbiri 168	. . . . . 6 |- ( A = -oo -> A < +oo )
22	21	necon3bi 2462	. . . . 5 |- ( -. A < +oo -> A =/= -oo )
23	22	necon2bi 2467	. . . 4 |- ( A = -oo -> -. -. A < +oo )
24	18, 23	2falsed 710	. . 3 |- ( A = -oo -> ( A = +oo <-> -. A < +oo ) )
25	7, 14, 24	3jaoi 1340	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A = +oo <-> -. A < +oo ) )
26	1, 25	sylbi 121	1 |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ w3o 1004   = wceq 1398   e. wcel 2203   class class class wbr 4109   RRcr 8126   +oocpnf 8305   -oocmnf 8306   RR*cxr 8307   < clt 8308

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-pre-ltirr 8239

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313

This theorem is referenced by:  npnflt  10148  xgepnf  10149  xrmaxiflemlub  11933