Theorem nltpnft 9936

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 9898	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 8120	. . . . 5 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2397	. . . 4 |- ( A e. RR -> -. A = +oo )
4		ltpnf 9902	. . . . 5 |- ( A e. RR -> A < +oo )
5		notnot 630	. . . . 5 |- ( A < +oo -> -. -. A < +oo )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. A < +oo )
7	3, 6	2falsed 704	. . 3 |- ( A e. RR -> ( A = +oo <-> -. A < +oo ) )
8		id 19	. . . 4 |- ( A = +oo -> A = +oo )
9		pnfxr 8125	. . . . . 6 |- +oo e. RR*
10		xrltnr 9901	. . . . . 6 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . 5 |- -. +oo < +oo
12		breq1 4047	. . . . 5 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 677	. . . 4 |- ( A = +oo -> -. A < +oo )
14	8, 13	2thd 175	. . 3 |- ( A = +oo -> ( A = +oo <-> -. A < +oo ) )
15		mnfnepnf 8128	. . . . . 6 |- -oo =/= +oo
16	15	neii 2378	. . . . 5 |- -. -oo = +oo
17		eqeq1 2212	. . . . 5 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
18	16, 17	mtbiri 677	. . . 4 |- ( A = -oo -> -. A = +oo )
19		mnfltpnf 9907	. . . . . . 7 |- -oo < +oo
20		breq1 4047	. . . . . . 7 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
21	19, 20	mpbiri 168	. . . . . 6 |- ( A = -oo -> A < +oo )
22	21	necon3bi 2426	. . . . 5 |- ( -. A < +oo -> A =/= -oo )
23	22	necon2bi 2431	. . . 4 |- ( A = -oo -> -. -. A < +oo )
24	18, 23	2falsed 704	. . 3 |- ( A = -oo -> ( A = +oo <-> -. A < +oo ) )
25	7, 14, 24	3jaoi 1316	. 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 980   = wceq 1373   e. wcel 2176   class class class wbr 4044   RRcr 7924   +oocpnf 8104   -oocmnf 8105   RR*cxr 8106   < clt 8107

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-pre-ltirr 8037

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112

This theorem is referenced by:  npnflt  9937  xgepnf  9938  xrmaxiflemlub  11559