Theorem nltpnft 9816

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 9778	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 8007	. . . . 5 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2368	. . . 4 |- ( A e. RR -> -. A = +oo )
4		ltpnf 9782	. . . . 5 |- ( A e. RR -> A < +oo )
5		notnot 629	. . . . 5 |- ( A < +oo -> -. -. A < +oo )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. A < +oo )
7	3, 6	2falsed 702	. . 3 |- ( A e. RR -> ( A = +oo <-> -. A < +oo ) )
8		id 19	. . . 4 |- ( A = +oo -> A = +oo )
9		pnfxr 8012	. . . . . 6 |- +oo e. RR*
10		xrltnr 9781	. . . . . 6 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . 5 |- -. +oo < +oo
12		breq1 4008	. . . . 5 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 675	. . . 4 |- ( A = +oo -> -. A < +oo )
14	8, 13	2thd 175	. . 3 |- ( A = +oo -> ( A = +oo <-> -. A < +oo ) )
15		mnfnepnf 8015	. . . . . 6 |- -oo =/= +oo
16	15	neii 2349	. . . . 5 |- -. -oo = +oo
17		eqeq1 2184	. . . . 5 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
18	16, 17	mtbiri 675	. . . 4 |- ( A = -oo -> -. A = +oo )
19		mnfltpnf 9787	. . . . . . 7 |- -oo < +oo
20		breq1 4008	. . . . . . 7 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
21	19, 20	mpbiri 168	. . . . . 6 |- ( A = -oo -> A < +oo )
22	21	necon3bi 2397	. . . . 5 |- ( -. A < +oo -> A =/= -oo )
23	22	necon2bi 2402	. . . 4 |- ( A = -oo -> -. -. A < +oo )
24	18, 23	2falsed 702	. . 3 |- ( A = -oo -> ( A = +oo <-> -. A < +oo ) )
25	7, 14, 24	3jaoi 1303	. 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 977   = wceq 1353   e. wcel 2148   class class class wbr 4005   RRcr 7812   +oocpnf 7991   -oocmnf 7992   RR*cxr 7993   < clt 7994

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-pre-ltirr 7925

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999

This theorem is referenced by:  npnflt  9817  xgepnf  9818  xrmaxiflemlub  11258