Theorem ngtmnft 9938

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 9897	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8120	. . . . 5 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2396	. . . 4 |- ( A e. RR -> -. A = -oo )
4		mnflt 9904	. . . . 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 8126	. . . . . 6 |- +oo =/= -oo
9		neeq1 2388	. . . . . 6 |- ( A = +oo -> ( A =/= -oo <-> +oo =/= -oo ) )
10	8, 9	mpbiri 168	. . . . 5 |- ( A = +oo -> A =/= -oo )
11	10	neneqd 2396	. . . 4 |- ( A = +oo -> -. A = -oo )
12		mnfltpnf 9906	. . . . . . 7 |- -oo < +oo
13		breq2 4047	. . . . . . 7 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
14	12, 13	mpbiri 168	. . . . . 6 |- ( A = +oo -> -oo < A )
15	14	necon3bi 2425	. . . . 5 |- ( -. -oo < A -> A =/= +oo )
16	15	necon2bi 2430	. . . 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 8128	. . . . . 6 |- -oo e. RR*
20		xrltnr 9900	. . . . . 6 |- ( -oo e. RR* -> -. -oo < -oo )
21	19, 20	ax-mp 5	. . . . 5 |- -. -oo < -oo
22		breq2 4047	. . . . 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 1315	. 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 1372   e. wcel 2175   =/= wne 2375   class class class wbr 4043   RRcr 7923   +oocpnf 8103   -oocmnf 8104   RR*cxr 8105   < clt 8106

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-pre-ltirr 8036

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-xp 4680  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111

This theorem is referenced by:  nmnfgt  9939  ge0nemnf  9945  xleaddadd  10008