Theorem xrltnsym 8996

Description: Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)

Assertion

Ref	Expression
xrltnsym	|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -. B < A ) )

Proof of Theorem xrltnsym

Step	Hyp	Ref	Expression
1		elxr 8980	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 8980	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		ltnsym 7316	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
4		rexr 7278	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
5		pnfnlt 8990	. . . . . . . 8 |- ( A e. RR* -> -. +oo < A )
6	4, 5	syl 14	. . . . . . 7 |- ( A e. RR -> -. +oo < A )
7	6	adantr 270	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> -. +oo < A )
8		breq1 3808	. . . . . . 7 |- ( B = +oo -> ( B < A <-> +oo < A ) )
9	8	adantl 271	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( B < A <-> +oo < A ) )
10	7, 9	mtbird 631	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -. B < A )
11	10	a1d 22	. . . 4 |- ( ( A e. RR /\ B = +oo ) -> ( A < B -> -. B < A ) )
12		nltmnf 8991	. . . . . . . 8 |- ( A e. RR* -> -. A < -oo )
13	4, 12	syl 14	. . . . . . 7 |- ( A e. RR -> -. A < -oo )
14	13	adantr 270	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
15		breq2 3809	. . . . . . 7 |- ( B = -oo -> ( A < B <-> A < -oo ) )
16	15	adantl 271	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
17	14, 16	mtbird 631	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
18	17	pm2.21d 582	. . . 4 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> -. B < A ) )
19	3, 11, 18	3jaodan 1238	. . 3 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
20		pnfnlt 8990	. . . . . . 7 |- ( B e. RR* -> -. +oo < B )
21	20	adantl 271	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
22		breq1 3808	. . . . . . 7 |- ( A = +oo -> ( A < B <-> +oo < B ) )
23	22	adantr 270	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
24	21, 23	mtbird 631	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -. A < B )
25	24	pm2.21d 582	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> -. B < A ) )
26	2, 25	sylan2br 282	. . 3 |- ( ( A = +oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
27		rexr 7278	. . . . . . . 8 |- ( B e. RR -> B e. RR* )
28		nltmnf 8991	. . . . . . . 8 |- ( B e. RR* -> -. B < -oo )
29	27, 28	syl 14	. . . . . . 7 |- ( B e. RR -> -. B < -oo )
30	29	adantl 271	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> -. B < -oo )
31		breq2 3809	. . . . . . 7 |- ( A = -oo -> ( B < A <-> B < -oo ) )
32	31	adantr 270	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> ( B < A <-> B < -oo ) )
33	30, 32	mtbird 631	. . . . 5 |- ( ( A = -oo /\ B e. RR ) -> -. B < A )
34	33	a1d 22	. . . 4 |- ( ( A = -oo /\ B e. RR ) -> ( A < B -> -. B < A ) )
35		mnfxr 7289	. . . . . . . 8 |- -oo e. RR*
36		pnfnlt 8990	. . . . . . . 8 |- ( -oo e. RR* -> -. +oo < -oo )
37	35, 36	ax-mp 7	. . . . . . 7 |- -. +oo < -oo
38		breq12 3810	. . . . . . 7 |- ( ( B = +oo /\ A = -oo ) -> ( B < A <-> +oo < -oo ) )
39	37, 38	mtbiri 633	. . . . . 6 |- ( ( B = +oo /\ A = -oo ) -> -. B < A )
40	39	ancoms 264	. . . . 5 |- ( ( A = -oo /\ B = +oo ) -> -. B < A )
41	40	a1d 22	. . . 4 |- ( ( A = -oo /\ B = +oo ) -> ( A < B -> -. B < A ) )
42		xrltnr 8983	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
43	35, 42	ax-mp 7	. . . . . 6 |- -. -oo < -oo
44		breq12 3810	. . . . . 6 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
45	43, 44	mtbiri 633	. . . . 5 |- ( ( A = -oo /\ B = -oo ) -> -. A < B )
46	45	pm2.21d 582	. . . 4 |- ( ( A = -oo /\ B = -oo ) -> ( A < B -> -. B < A ) )
47	34, 41, 46	3jaodan 1238	. . 3 |- ( ( A = -oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
48	19, 26, 47	3jaoian 1237	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
49	1, 2, 48	syl2anb 285	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -. B < A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ w3o 919   = wceq 1285   e. wcel 1434   class class class wbr 3805   RRcr 7094   +oocpnf 7264   -oocmnf 7265   RR*cxr 7266   < clt 7267

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-pre-ltirr 7202  ax-pre-lttrn 7204

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272

This theorem is referenced by:  xrltnsym2  8997  xrltle  9001