Theorem ngtmnft 10009

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 9968	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8191	. . . . 5 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2421	. . . 4 |- ( A e. RR -> -. A = -oo )
4		mnflt 9975	. . . . 5 |- ( A e. RR -> -oo < A )
5		notnot 632	. . . . 5 |- ( -oo < A -> -. -. -oo < A )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. -oo < A )
7	3, 6	2falsed 707	. . 3 |- ( A e. RR -> ( A = -oo <-> -. -oo < A ) )
8		pnfnemnf 8197	. . . . . 6 |- +oo =/= -oo
9		neeq1 2413	. . . . . 6 |- ( A = +oo -> ( A =/= -oo <-> +oo =/= -oo ) )
10	8, 9	mpbiri 168	. . . . 5 |- ( A = +oo -> A =/= -oo )
11	10	neneqd 2421	. . . 4 |- ( A = +oo -> -. A = -oo )
12		mnfltpnf 9977	. . . . . . 7 |- -oo < +oo
13		breq2 4086	. . . . . . 7 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
14	12, 13	mpbiri 168	. . . . . 6 |- ( A = +oo -> -oo < A )
15	14	necon3bi 2450	. . . . 5 |- ( -. -oo < A -> A =/= +oo )
16	15	necon2bi 2455	. . . 4 |- ( A = +oo -> -. -. -oo < A )
17	11, 16	2falsed 707	. . 3 |- ( A = +oo -> ( A = -oo <-> -. -oo < A ) )
18		id 19	. . . 4 |- ( A = -oo -> A = -oo )
19		mnfxr 8199	. . . . . 6 |- -oo e. RR*
20		xrltnr 9971	. . . . . 6 |- ( -oo e. RR* -> -. -oo < -oo )
21	19, 20	ax-mp 5	. . . . 5 |- -. -oo < -oo
22		breq2 4086	. . . . 5 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
23	21, 22	mtbiri 679	. . . 4 |- ( A = -oo -> -. -oo < A )
24	18, 23	2thd 175	. . 3 |- ( A = -oo -> ( A = -oo <-> -. -oo < A ) )
25	7, 17, 24	3jaoi 1337	. 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 1001   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4082   RRcr 7994   +oocpnf 8174   -oocmnf 8175   RR*cxr 8176   < clt 8177

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-pre-ltirr 8107

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182

This theorem is referenced by:  nmnfgt  10010  ge0nemnf  10016  xleaddadd  10079