Theorem ngtmnft 9816

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 9775	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8005	. . . . 5 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2368	. . . 4 |- ( A e. RR -> -. A = -oo )
4		mnflt 9782	. . . . 5 |- ( A e. RR -> -oo < A )
5		notnot 629	. . . . 5 |- ( -oo < A -> -. -. -oo < A )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. -oo < A )
7	3, 6	2falsed 702	. . 3 |- ( A e. RR -> ( A = -oo <-> -. -oo < A ) )
8		pnfnemnf 8011	. . . . . 6 |- +oo =/= -oo
9		neeq1 2360	. . . . . 6 |- ( A = +oo -> ( A =/= -oo <-> +oo =/= -oo ) )
10	8, 9	mpbiri 168	. . . . 5 |- ( A = +oo -> A =/= -oo )
11	10	neneqd 2368	. . . 4 |- ( A = +oo -> -. A = -oo )
12		mnfltpnf 9784	. . . . . . 7 |- -oo < +oo
13		breq2 4007	. . . . . . 7 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
14	12, 13	mpbiri 168	. . . . . 6 |- ( A = +oo -> -oo < A )
15	14	necon3bi 2397	. . . . 5 |- ( -. -oo < A -> A =/= +oo )
16	15	necon2bi 2402	. . . 4 |- ( A = +oo -> -. -. -oo < A )
17	11, 16	2falsed 702	. . 3 |- ( A = +oo -> ( A = -oo <-> -. -oo < A ) )
18		id 19	. . . 4 |- ( A = -oo -> A = -oo )
19		mnfxr 8013	. . . . . 6 |- -oo e. RR*
20		xrltnr 9778	. . . . . 6 |- ( -oo e. RR* -> -. -oo < -oo )
21	19, 20	ax-mp 5	. . . . 5 |- -. -oo < -oo
22		breq2 4007	. . . . 5 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
23	21, 22	mtbiri 675	. . . 4 |- ( A = -oo -> -. -oo < A )
24	18, 23	2thd 175	. . 3 |- ( A = -oo -> ( A = -oo <-> -. -oo < A ) )
25	7, 17, 24	3jaoi 1303	. 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 977   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4003   RRcr 7809   +oocpnf 7988   -oocmnf 7989   RR*cxr 7990   < clt 7991

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltirr 7922

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996

This theorem is referenced by:  nmnfgt  9817  ge0nemnf  9823  xleaddadd  9886