Theorem xrltnsym 9989

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 9972	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9972	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		ltnsym 8232	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
4		rexr 8192	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
5		pnfnlt 9983	. . . . . . . 8 |- ( A e. RR* -> -. +oo < A )
6	4, 5	syl 14	. . . . . . 7 |- ( A e. RR -> -. +oo < A )
7	6	adantr 276	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> -. +oo < A )
8		breq1 4086	. . . . . . 7 |- ( B = +oo -> ( B < A <-> +oo < A ) )
9	8	adantl 277	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( B < A <-> +oo < A ) )
10	7, 9	mtbird 677	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -. B < A )
11	10	a1d 22	. . . 4 |- ( ( A e. RR /\ B = +oo ) -> ( A < B -> -. B < A ) )
12		nltmnf 9984	. . . . . . . 8 |- ( A e. RR* -> -. A < -oo )
13	4, 12	syl 14	. . . . . . 7 |- ( A e. RR -> -. A < -oo )
14	13	adantr 276	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
15		breq2 4087	. . . . . . 7 |- ( B = -oo -> ( A < B <-> A < -oo ) )
16	15	adantl 277	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
17	14, 16	mtbird 677	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
18	17	pm2.21d 622	. . . 4 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> -. B < A ) )
19	3, 11, 18	3jaodan 1340	. . 3 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
20		pnfnlt 9983	. . . . . . 7 |- ( B e. RR* -> -. +oo < B )
21	20	adantl 277	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
22		breq1 4086	. . . . . . 7 |- ( A = +oo -> ( A < B <-> +oo < B ) )
23	22	adantr 276	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
24	21, 23	mtbird 677	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -. A < B )
25	24	pm2.21d 622	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> -. B < A ) )
26	2, 25	sylan2br 288	. . 3 |- ( ( A = +oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
27		rexr 8192	. . . . . . . 8 |- ( B e. RR -> B e. RR* )
28		nltmnf 9984	. . . . . . . 8 |- ( B e. RR* -> -. B < -oo )
29	27, 28	syl 14	. . . . . . 7 |- ( B e. RR -> -. B < -oo )
30	29	adantl 277	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> -. B < -oo )
31		breq2 4087	. . . . . . 7 |- ( A = -oo -> ( B < A <-> B < -oo ) )
32	31	adantr 276	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> ( B < A <-> B < -oo ) )
33	30, 32	mtbird 677	. . . . 5 |- ( ( A = -oo /\ B e. RR ) -> -. B < A )
34	33	a1d 22	. . . 4 |- ( ( A = -oo /\ B e. RR ) -> ( A < B -> -. B < A ) )
35		mnfxr 8203	. . . . . . . 8 |- -oo e. RR*
36		pnfnlt 9983	. . . . . . . 8 |- ( -oo e. RR* -> -. +oo < -oo )
37	35, 36	ax-mp 5	. . . . . . 7 |- -. +oo < -oo
38		breq12 4088	. . . . . . 7 |- ( ( B = +oo /\ A = -oo ) -> ( B < A <-> +oo < -oo ) )
39	37, 38	mtbiri 679	. . . . . 6 |- ( ( B = +oo /\ A = -oo ) -> -. B < A )
40	39	ancoms 268	. . . . 5 |- ( ( A = -oo /\ B = +oo ) -> -. B < A )
41	40	a1d 22	. . . 4 |- ( ( A = -oo /\ B = +oo ) -> ( A < B -> -. B < A ) )
42		xrltnr 9975	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
43	35, 42	ax-mp 5	. . . . . 6 |- -. -oo < -oo
44		breq12 4088	. . . . . 6 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
45	43, 44	mtbiri 679	. . . . 5 |- ( ( A = -oo /\ B = -oo ) -> -. A < B )
46	45	pm2.21d 622	. . . 4 |- ( ( A = -oo /\ B = -oo ) -> ( A < B -> -. B < A ) )
47	34, 41, 46	3jaodan 1340	. . 3 |- ( ( A = -oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
48	19, 26, 47	3jaoian 1339	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
49	1, 2, 48	syl2anb 291	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -. B < A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1001   = wceq 1395   e. wcel 2200   class class class wbr 4083   RRcr 7998   +oocpnf 8178   -oocmnf 8179   RR*cxr 8180   < clt 8181

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-pre-ltirr 8111  ax-pre-lttrn 8113

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186

This theorem is referenced by:  xrltnsym2  9990  xrltle  9994