Theorem ngtmnft 9892

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 9851	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8075	. . . . 5 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2388	. . . 4 |- ( A e. RR -> -. A = -oo )
4		mnflt 9858	. . . . 5 |- ( A e. RR -> -oo < A )
5		notnot 630	. . . . 5 |- ( -oo < A -> -. -. -oo < A )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. -oo < A )
7	3, 6	2falsed 703	. . 3 |- ( A e. RR -> ( A = -oo <-> -. -oo < A ) )
8		pnfnemnf 8081	. . . . . 6 |- +oo =/= -oo
9		neeq1 2380	. . . . . 6 |- ( A = +oo -> ( A =/= -oo <-> +oo =/= -oo ) )
10	8, 9	mpbiri 168	. . . . 5 |- ( A = +oo -> A =/= -oo )
11	10	neneqd 2388	. . . 4 |- ( A = +oo -> -. A = -oo )
12		mnfltpnf 9860	. . . . . . 7 |- -oo < +oo
13		breq2 4037	. . . . . . 7 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
14	12, 13	mpbiri 168	. . . . . 6 |- ( A = +oo -> -oo < A )
15	14	necon3bi 2417	. . . . 5 |- ( -. -oo < A -> A =/= +oo )
16	15	necon2bi 2422	. . . 4 |- ( A = +oo -> -. -. -oo < A )
17	11, 16	2falsed 703	. . 3 |- ( A = +oo -> ( A = -oo <-> -. -oo < A ) )
18		id 19	. . . 4 |- ( A = -oo -> A = -oo )
19		mnfxr 8083	. . . . . 6 |- -oo e. RR*
20		xrltnr 9854	. . . . . 6 |- ( -oo e. RR* -> -. -oo < -oo )
21	19, 20	ax-mp 5	. . . . 5 |- -. -oo < -oo
22		breq2 4037	. . . . 5 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
23	21, 22	mtbiri 676	. . . 4 |- ( A = -oo -> -. -oo < A )
24	18, 23	2thd 175	. . 3 |- ( A = -oo -> ( A = -oo <-> -. -oo < A ) )
25	7, 17, 24	3jaoi 1314	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A = -oo <-> -. -oo < A ) )
26	1, 25	sylbi 121	1 |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ w3o 979   = wceq 1364   e. wcel 2167   =/= wne 2367   class class class wbr 4033   RRcr 7878   +oocpnf 8058   -oocmnf 8059   RR*cxr 8060   < clt 8061

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-pre-ltirr 7991

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066

This theorem is referenced by:  nmnfgt  9893  ge0nemnf  9899  xleaddadd  9962