Theorem ngtmnft 9721

Description: An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)

Assertion

Ref	Expression
ngtmnft	|- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )

Proof of Theorem ngtmnft

Step	Hyp	Ref	Expression
1		elxr 9683	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 7926	. . . . 5 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2348	. . . 4 |- ( A e. RR -> -. A = -oo )
4		mnflt 9690	. . . . 5 |- ( A e. RR -> -oo < A )
5		notnot 619	. . . . 5 |- ( -oo < A -> -. -. -oo < A )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. -oo < A )
7	3, 6	2falsed 692	. . 3 |- ( A e. RR -> ( A = -oo <-> -. -oo < A ) )
8		pnfnemnf 7932	. . . . . 6 |- +oo =/= -oo
9		neeq1 2340	. . . . . 6 |- ( A = +oo -> ( A =/= -oo <-> +oo =/= -oo ) )
10	8, 9	mpbiri 167	. . . . 5 |- ( A = +oo -> A =/= -oo )
11	10	neneqd 2348	. . . 4 |- ( A = +oo -> -. A = -oo )
12		mnfltpnf 9692	. . . . . . 7 |- -oo < +oo
13		breq2 3969	. . . . . . 7 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
14	12, 13	mpbiri 167	. . . . . 6 |- ( A = +oo -> -oo < A )
15	14	necon3bi 2377	. . . . 5 |- ( -. -oo < A -> A =/= +oo )
16	15	necon2bi 2382	. . . 4 |- ( A = +oo -> -. -. -oo < A )
17	11, 16	2falsed 692	. . 3 |- ( A = +oo -> ( A = -oo <-> -. -oo < A ) )
18		id 19	. . . 4 |- ( A = -oo -> A = -oo )
19		mnfxr 7934	. . . . . 6 |- -oo e. RR*
20		xrltnr 9686	. . . . . 6 |- ( -oo e. RR* -> -. -oo < -oo )
21	19, 20	ax-mp 5	. . . . 5 |- -. -oo < -oo
22		breq2 3969	. . . . 5 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
23	21, 22	mtbiri 665	. . . 4 |- ( A = -oo -> -. -oo < A )
24	18, 23	2thd 174	. . 3 |- ( A = -oo -> ( A = -oo <-> -. -oo < A ) )
25	7, 17, 24	3jaoi 1285	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A = -oo <-> -. -oo < A ) )
26	1, 25	sylbi 120	1 |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ w3o 962   = wceq 1335   e. wcel 2128   =/= wne 2327   class class class wbr 3965   RRcr 7731   +oocpnf 7909   -oocmnf 7910   RR*cxr 7911   < clt 7912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-pre-ltirr 7844

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917

This theorem is referenced by:  nmnfgt  9722  ge0nemnf  9728  xleaddadd  9791