Theorem ngtmnft 10169

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 10128	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8338	. . . . 5 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2435	. . . 4 |- ( A e. RR -> -. A = -oo )
4		mnflt 10135	. . . . 5 |- ( A e. RR -> -oo < A )
5		notnot 634	. . . . 5 |- ( -oo < A -> -. -. -oo < A )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. -oo < A )
7	3, 6	2falsed 710	. . 3 |- ( A e. RR -> ( A = -oo <-> -. -oo < A ) )
8		pnfnemnf 8344	. . . . . 6 |- +oo =/= -oo
9		neeq1 2427	. . . . . 6 |- ( A = +oo -> ( A =/= -oo <-> +oo =/= -oo ) )
10	8, 9	mpbiri 168	. . . . 5 |- ( A = +oo -> A =/= -oo )
11	10	neneqd 2435	. . . 4 |- ( A = +oo -> -. A = -oo )
12		mnfltpnf 10137	. . . . . . 7 |- -oo < +oo
13		breq2 4118	. . . . . . 7 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
14	12, 13	mpbiri 168	. . . . . 6 |- ( A = +oo -> -oo < A )
15	14	necon3bi 2464	. . . . 5 |- ( -. -oo < A -> A =/= +oo )
16	15	necon2bi 2469	. . . 4 |- ( A = +oo -> -. -. -oo < A )
17	11, 16	2falsed 710	. . 3 |- ( A = +oo -> ( A = -oo <-> -. -oo < A ) )
18		id 19	. . . 4 |- ( A = -oo -> A = -oo )
19		mnfxr 8346	. . . . . 6 |- -oo e. RR*
20		xrltnr 10131	. . . . . 6 |- ( -oo e. RR* -> -. -oo < -oo )
21	19, 20	ax-mp 5	. . . . 5 |- -. -oo < -oo
22		breq2 4118	. . . . 5 |- ( A = -oo -> ( -oo < A <-> -oo < -oo ) )
23	21, 22	mtbiri 682	. . . 4 |- ( A = -oo -> -. -oo < A )
24	18, 23	2thd 175	. . 3 |- ( A = -oo -> ( A = -oo <-> -. -oo < A ) )
25	7, 17, 24	3jaoi 1340	. 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 1004   = wceq 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4114   RRcr 8142   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   < clt 8324

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-pre-ltirr 8255

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329

This theorem is referenced by:  nmnfgt  10170  ge0nemnf  10176  xleaddadd  10239