Theorem xrltnsym 8714

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 8696	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 8696	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		ltnsym 7104	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
4		rexr 7071	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
5		pnfnlt 8708	. . . . . . . 8 |- ( A e. RR* -> -. +oo < A )
6	4, 5	syl 14	. . . . . . 7 |- ( A e. RR -> -. +oo < A )
7	6	adantr 261	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> -. +oo < A )
8		breq1 3767	. . . . . . 7 |- ( B = +oo -> ( B < A <-> +oo < A ) )
9	8	adantl 262	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( B < A <-> +oo < A ) )
10	7, 9	mtbird 598	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -. B < A )
11	10	a1d 22	. . . 4 |- ( ( A e. RR /\ B = +oo ) -> ( A < B -> -. B < A ) )
12		nltmnf 8709	. . . . . . . 8 |- ( A e. RR* -> -. A < -oo )
13	4, 12	syl 14	. . . . . . 7 |- ( A e. RR -> -. A < -oo )
14	13	adantr 261	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
15		breq2 3768	. . . . . . 7 |- ( B = -oo -> ( A < B <-> A < -oo ) )
16	15	adantl 262	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
17	14, 16	mtbird 598	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
18	17	pm2.21d 549	. . . 4 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> -. B < A ) )
19	3, 11, 18	3jaodan 1201	. . 3 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
20		pnfnlt 8708	. . . . . . 7 |- ( B e. RR* -> -. +oo < B )
21	20	adantl 262	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
22		breq1 3767	. . . . . . 7 |- ( A = +oo -> ( A < B <-> +oo < B ) )
23	22	adantr 261	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
24	21, 23	mtbird 598	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -. A < B )
25	24	pm2.21d 549	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> -. B < A ) )
26	2, 25	sylan2br 272	. . 3 |- ( ( A = +oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
27		rexr 7071	. . . . . . . 8 |- ( B e. RR -> B e. RR* )
28		nltmnf 8709	. . . . . . . 8 |- ( B e. RR* -> -. B < -oo )
29	27, 28	syl 14	. . . . . . 7 |- ( B e. RR -> -. B < -oo )
30	29	adantl 262	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> -. B < -oo )
31		breq2 3768	. . . . . . 7 |- ( A = -oo -> ( B < A <-> B < -oo ) )
32	31	adantr 261	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> ( B < A <-> B < -oo ) )
33	30, 32	mtbird 598	. . . . 5 |- ( ( A = -oo /\ B e. RR ) -> -. B < A )
34	33	a1d 22	. . . 4 |- ( ( A = -oo /\ B e. RR ) -> ( A < B -> -. B < A ) )
35		mnfxr 8694	. . . . . . . 8 |- -oo e. RR*
36		pnfnlt 8708	. . . . . . . 8 |- ( -oo e. RR* -> -. +oo < -oo )
37	35, 36	ax-mp 7	. . . . . . 7 |- -. +oo < -oo
38		breq12 3769	. . . . . . 7 |- ( ( B = +oo /\ A = -oo ) -> ( B < A <-> +oo < -oo ) )
39	37, 38	mtbiri 600	. . . . . 6 |- ( ( B = +oo /\ A = -oo ) -> -. B < A )
40	39	ancoms 255	. . . . 5 |- ( ( A = -oo /\ B = +oo ) -> -. B < A )
41	40	a1d 22	. . . 4 |- ( ( A = -oo /\ B = +oo ) -> ( A < B -> -. B < A ) )
42		xrltnr 8701	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
43	35, 42	ax-mp 7	. . . . . 6 |- -. -oo < -oo
44		breq12 3769	. . . . . 6 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
45	43, 44	mtbiri 600	. . . . 5 |- ( ( A = -oo /\ B = -oo ) -> -. A < B )
46	45	pm2.21d 549	. . . 4 |- ( ( A = -oo /\ B = -oo ) -> ( A < B -> -. B < A ) )
47	34, 41, 46	3jaodan 1201	. . 3 |- ( ( A = -oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
48	19, 26, 47	3jaoian 1200	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
49	1, 2, 48	syl2anb 275	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -. B < A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ w3o 884   = wceq 1243   e. wcel 1393   class class class wbr 3764   RRcr 6888   +oocpnf 7057   -oocmnf 7058   RR*cxr 7059   < clt 7060

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996  ax-pre-lttrn 6998

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065

This theorem is referenced by:  xrltnsym2  8715  xrltle  8719