Theorem nltpnft 10216

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 10178	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 8373	. . . . 5 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2441	. . . 4 |- ( A e. RR -> -. A = +oo )
4		ltpnf 10182	. . . . 5 |- ( A e. RR -> A < +oo )
5		notnot 638	. . . . 5 |- ( A < +oo -> -. -. A < +oo )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. A < +oo )
7	3, 6	2falsed 714	. . 3 |- ( A e. RR -> ( A = +oo <-> -. A < +oo ) )
8		id 19	. . . 4 |- ( A = +oo -> A = +oo )
9		pnfxr 8378	. . . . . 6 |- +oo e. RR*
10		xrltnr 10181	. . . . . 6 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . 5 |- -. +oo < +oo
12		breq1 4133	. . . . 5 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 686	. . . 4 |- ( A = +oo -> -. A < +oo )
14	8, 13	2thd 175	. . 3 |- ( A = +oo -> ( A = +oo <-> -. A < +oo ) )
15		mnfnepnf 8381	. . . . . 6 |- -oo =/= +oo
16	15	neii 2422	. . . . 5 |- -. -oo = +oo
17		eqeq1 2245	. . . . 5 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
18	16, 17	mtbiri 686	. . . 4 |- ( A = -oo -> -. A = +oo )
19		mnfltpnf 10187	. . . . . . 7 |- -oo < +oo
20		breq1 4133	. . . . . . 7 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
21	19, 20	mpbiri 168	. . . . . 6 |- ( A = -oo -> A < +oo )
22	21	necon3bi 2470	. . . . 5 |- ( -. A < +oo -> A =/= -oo )
23	22	necon2bi 2475	. . . 4 |- ( A = -oo -> -. -. A < +oo )
24	18, 23	2falsed 714	. . 3 |- ( A = -oo -> ( A = +oo <-> -. A < +oo ) )
25	7, 14, 24	3jaoi 1344	. 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 ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   <-> wb 105   \/ w3o 1008   = wceq 1402   e. wcel 2209   class class class wbr 4130   RRcr 8178   +oocpnf 8357   -oocmnf 8358   RR*cxr 8359   < clt 8360

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365

This theorem is used by:  npnflt  10217  xgepnf  10218  xrmaxiflemlub  12014