Theorem ngtmnft 9278

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 9245	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 7534	. . . . 5 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2276	. . . 4 |- ( A e. RR -> -. A = -oo )
4		mnflt 9251	. . . . 5 |- ( A e. RR -> -oo < A )
5		notnot 594	. . . . 5 |- ( -oo < A -> -. -. -oo < A )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. -oo < A )
7	3, 6	2falsed 653	. . 3 |- ( A e. RR -> ( A = -oo <-> -. -oo < A ) )
8		pnfnemnf 7540	. . . . . 6 |- +oo =/= -oo
9		neeq1 2268	. . . . . 6 |- ( A = +oo -> ( A =/= -oo <-> +oo =/= -oo ) )
10	8, 9	mpbiri 166	. . . . 5 |- ( A = +oo -> A =/= -oo )
11	10	neneqd 2276	. . . 4 |- ( A = +oo -> -. A = -oo )
12		mnfltpnf 9253	. . . . . . 7 |- -oo < +oo
13		breq2 3849	. . . . . . 7 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
14	12, 13	mpbiri 166	. . . . . 6 |- ( A = +oo -> -oo < A )
15	14	necon3bi 2305	. . . . 5 |- ( -. -oo < A -> A =/= +oo )
16	15	necon2bi 2310	. . . 4 |- ( A = +oo -> -. -. -oo < A )
17	11, 16	2falsed 653	. . 3 |- ( A = +oo -> ( A = -oo <-> -. -oo < A ) )
18		id 19	. . . 4 |- ( A = -oo -> A = -oo )
19		mnfxr 7542	. . . . . 6 |- -oo e. RR*
20		xrltnr 9248	. . . . . 6 |- ( -oo e. RR* -> -. -oo < -oo )
21	19, 20	ax-mp 7	. . . . 5 |- -. -oo < -oo
22		breq2 3849	. . . . 5 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
23	21, 22	mtbiri 635	. . . 4 |- ( A = -oo -> -. -oo < A )
24	18, 23	2thd 173	. . 3 |- ( A = -oo -> ( A = -oo <-> -. -oo < A ) )
25	7, 17, 24	3jaoi 1239	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A = -oo <-> -. -oo < A ) )
26	1, 25	sylbi 119	1 |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ w3o 923   = wceq 1289   e. wcel 1438   =/= wne 2255   class class class wbr 3845   RRcr 7347   +oocpnf 7517   -oocmnf 7518   RR*cxr 7519   < clt 7520

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-pre-ltirr 7455

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525

This theorem is referenced by:  ge0nemnf  9284