Theorem xrltnsym 9579

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 9563	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9563	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		ltnsym 7850	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
4		rexr 7811	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
5		pnfnlt 9573	. . . . . . . 8 |- ( A e. RR* -> -. +oo < A )
6	4, 5	syl 14	. . . . . . 7 |- ( A e. RR -> -. +oo < A )
7	6	adantr 274	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> -. +oo < A )
8		breq1 3932	. . . . . . 7 |- ( B = +oo -> ( B < A <-> +oo < A ) )
9	8	adantl 275	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( B < A <-> +oo < A ) )
10	7, 9	mtbird 662	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -. B < A )
11	10	a1d 22	. . . 4 |- ( ( A e. RR /\ B = +oo ) -> ( A < B -> -. B < A ) )
12		nltmnf 9574	. . . . . . . 8 |- ( A e. RR* -> -. A < -oo )
13	4, 12	syl 14	. . . . . . 7 |- ( A e. RR -> -. A < -oo )
14	13	adantr 274	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
15		breq2 3933	. . . . . . 7 |- ( B = -oo -> ( A < B <-> A < -oo ) )
16	15	adantl 275	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
17	14, 16	mtbird 662	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
18	17	pm2.21d 608	. . . 4 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> -. B < A ) )
19	3, 11, 18	3jaodan 1284	. . 3 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
20		pnfnlt 9573	. . . . . . 7 |- ( B e. RR* -> -. +oo < B )
21	20	adantl 275	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
22		breq1 3932	. . . . . . 7 |- ( A = +oo -> ( A < B <-> +oo < B ) )
23	22	adantr 274	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
24	21, 23	mtbird 662	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -. A < B )
25	24	pm2.21d 608	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> -. B < A ) )
26	2, 25	sylan2br 286	. . 3 |- ( ( A = +oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
27		rexr 7811	. . . . . . . 8 |- ( B e. RR -> B e. RR* )
28		nltmnf 9574	. . . . . . . 8 |- ( B e. RR* -> -. B < -oo )
29	27, 28	syl 14	. . . . . . 7 |- ( B e. RR -> -. B < -oo )
30	29	adantl 275	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> -. B < -oo )
31		breq2 3933	. . . . . . 7 |- ( A = -oo -> ( B < A <-> B < -oo ) )
32	31	adantr 274	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> ( B < A <-> B < -oo ) )
33	30, 32	mtbird 662	. . . . 5 |- ( ( A = -oo /\ B e. RR ) -> -. B < A )
34	33	a1d 22	. . . 4 |- ( ( A = -oo /\ B e. RR ) -> ( A < B -> -. B < A ) )
35		mnfxr 7822	. . . . . . . 8 |- -oo e. RR*
36		pnfnlt 9573	. . . . . . . 8 |- ( -oo e. RR* -> -. +oo < -oo )
37	35, 36	ax-mp 5	. . . . . . 7 |- -. +oo < -oo
38		breq12 3934	. . . . . . 7 |- ( ( B = +oo /\ A = -oo ) -> ( B < A <-> +oo < -oo ) )
39	37, 38	mtbiri 664	. . . . . 6 |- ( ( B = +oo /\ A = -oo ) -> -. B < A )
40	39	ancoms 266	. . . . 5 |- ( ( A = -oo /\ B = +oo ) -> -. B < A )
41	40	a1d 22	. . . 4 |- ( ( A = -oo /\ B = +oo ) -> ( A < B -> -. B < A ) )
42		xrltnr 9566	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
43	35, 42	ax-mp 5	. . . . . 6 |- -. -oo < -oo
44		breq12 3934	. . . . . 6 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
45	43, 44	mtbiri 664	. . . . 5 |- ( ( A = -oo /\ B = -oo ) -> -. A < B )
46	45	pm2.21d 608	. . . 4 |- ( ( A = -oo /\ B = -oo ) -> ( A < B -> -. B < A ) )
47	34, 41, 46	3jaodan 1284	. . 3 |- ( ( A = -oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
48	19, 26, 47	3jaoian 1283	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
49	1, 2, 48	syl2anb 289	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -. B < A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 961   = wceq 1331   e. wcel 1480   class class class wbr 3929   RRcr 7619   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   < clt 7800

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732  ax-pre-lttrn 7734

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805

This theorem is referenced by:  xrltnsym2  9580  xrltle  9584