Theorem nltpnft 8730

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 8696	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 7073	. . . . 5 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2226	. . . 4 |- ( A e. RR -> -. A = +oo )
4		ltpnf 8702	. . . . 5 |- ( A e. RR -> A < +oo )
5		notnot 559	. . . . 5 |- ( A < +oo -> -. -. A < +oo )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. A < +oo )
7	3, 6	2falsed 618	. . 3 |- ( A e. RR -> ( A = +oo <-> -. A < +oo ) )
8		id 19	. . . 4 |- ( A = +oo -> A = +oo )
9		pnfxr 8692	. . . . . 6 |- +oo e. RR*
10		xrltnr 8701	. . . . . 6 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 7	. . . . 5 |- -. +oo < +oo
12		breq1 3767	. . . . 5 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 600	. . . 4 |- ( A = +oo -> -. A < +oo )
14	8, 13	2thd 164	. . 3 |- ( A = +oo -> ( A = +oo <-> -. A < +oo ) )
15		mnfnepnf 8698	. . . . . 6 |- -oo =/= +oo
16	15	neii 2208	. . . . 5 |- -. -oo = +oo
17		eqeq1 2046	. . . . 5 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
18	16, 17	mtbiri 600	. . . 4 |- ( A = -oo -> -. A = +oo )
19		mnfltpnf 8706	. . . . . . 7 |- -oo < +oo
20		breq1 3767	. . . . . . 7 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
21	19, 20	mpbiri 157	. . . . . 6 |- ( A = -oo -> A < +oo )
22	21	necon3bi 2255	. . . . 5 |- ( -. A < +oo -> A =/= -oo )
23	22	necon2bi 2260	. . . 4 |- ( A = -oo -> -. -. A < +oo )
24	18, 23	2falsed 618	. . 3 |- ( A = -oo -> ( A = +oo <-> -. A < +oo ) )
25	7, 14, 24	3jaoi 1198	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A = +oo <-> -. A < +oo ) )
26	1, 25	sylbi 114	1 |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   \/ w3o 884   = wceq 1243   e. wcel 1393   class class class wbr 3764   RRcr 6888   +oocpnf 7057   -oocmnf 7058   RR*cxr 7059   < clt 7060

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065

This theorem is referenced by: (None)