Theorem ngtmnft 9774

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 9733	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 7968	. . . . 5 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2361	. . . 4 |- ( A e. RR -> -. A = -oo )
4		mnflt 9740	. . . . 5 |- ( A e. RR -> -oo < A )
5		notnot 624	. . . . 5 |- ( -oo < A -> -. -. -oo < A )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. -oo < A )
7	3, 6	2falsed 697	. . 3 |- ( A e. RR -> ( A = -oo <-> -. -oo < A ) )
8		pnfnemnf 7974	. . . . . 6 |- +oo =/= -oo
9		neeq1 2353	. . . . . 6 |- ( A = +oo -> ( A =/= -oo <-> +oo =/= -oo ) )
10	8, 9	mpbiri 167	. . . . 5 |- ( A = +oo -> A =/= -oo )
11	10	neneqd 2361	. . . 4 |- ( A = +oo -> -. A = -oo )
12		mnfltpnf 9742	. . . . . . 7 |- -oo < +oo
13		breq2 3993	. . . . . . 7 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
14	12, 13	mpbiri 167	. . . . . 6 |- ( A = +oo -> -oo < A )
15	14	necon3bi 2390	. . . . 5 |- ( -. -oo < A -> A =/= +oo )
16	15	necon2bi 2395	. . . 4 |- ( A = +oo -> -. -. -oo < A )
17	11, 16	2falsed 697	. . 3 |- ( A = +oo -> ( A = -oo <-> -. -oo < A ) )
18		id 19	. . . 4 |- ( A = -oo -> A = -oo )
19		mnfxr 7976	. . . . . 6 |- -oo e. RR*
20		xrltnr 9736	. . . . . 6 |- ( -oo e. RR* -> -. -oo < -oo )
21	19, 20	ax-mp 5	. . . . 5 |- -. -oo < -oo
22		breq2 3993	. . . . 5 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
23	21, 22	mtbiri 670	. . . 4 |- ( A = -oo -> -. -oo < A )
24	18, 23	2thd 174	. . 3 |- ( A = -oo -> ( A = -oo <-> -. -oo < A ) )
25	7, 17, 24	3jaoi 1298	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A = -oo <-> -. -oo < A ) )
26	1, 25	sylbi 120	1 |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ w3o 972   = wceq 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3989   RRcr 7773   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953   < clt 7954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959

This theorem is referenced by:  nmnfgt  9775  ge0nemnf  9781  xleaddadd  9844