Theorem xrltnsym 10195

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 10178	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 10178	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		ltnsym 8411	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
4		rexr 8371	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
5		pnfnlt 10189	. . . . . . . 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 4133	. . . . . . 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 684	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -. B < A )
11	10	a1d 22	. . . 4 |- ( ( A e. RR /\ B = +oo ) -> ( A < B -> -. B < A ) )
12		nltmnf 10190	. . . . . . . 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 4134	. . . . . . 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 684	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
18	17	pm2.21d 628	. . . 4 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> -. B < A ) )
19	3, 11, 18	3jaodan 1347	. . 3 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
20		pnfnlt 10189	. . . . . . 7 |- ( B e. RR* -> -. +oo < B )
21	20	adantl 277	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
22		breq1 4133	. . . . . . 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 684	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -. A < B )
25	24	pm2.21d 628	. . . 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 8371	. . . . . . . 8 |- ( B e. RR -> B e. RR* )
28		nltmnf 10190	. . . . . . . 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 4134	. . . . . . 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 684	. . . . 5 |- ( ( A = -oo /\ B e. RR ) -> -. B < A )
34	33	a1d 22	. . . 4 |- ( ( A = -oo /\ B e. RR ) -> ( A < B -> -. B < A ) )
35		mnfxr 8382	. . . . . . . 8 |- -oo e. RR*
36		pnfnlt 10189	. . . . . . . 8 |- ( -oo e. RR* -> -. +oo < -oo )
37	35, 36	ax-mp 5	. . . . . . 7 |- -. +oo < -oo
38		breq12 4135	. . . . . . 7 |- ( ( B = +oo /\ A = -oo ) -> ( B < A <-> +oo < -oo ) )
39	37, 38	mtbiri 686	. . . . . 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 10181	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
43	35, 42	ax-mp 5	. . . . . 6 |- -. -oo < -oo
44		breq12 4135	. . . . . 6 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
45	43, 44	mtbiri 686	. . . . 5 |- ( ( A = -oo /\ B = -oo ) -> -. A < B )
46	45	pm2.21d 628	. . . 4 |- ( ( A = -oo /\ B = -oo ) -> ( A < B -> -. B < A ) )
47	34, 41, 46	3jaodan 1347	. . 3 |- ( ( A = -oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
48	19, 26, 47	3jaoian 1346	. 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 ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1008   = wceq 1402   e. wcel 2209   class class class wbr 4130   RRcr 8178   +oocpnf 8357   -oocmnf 8358   RR*cxr 8359   < clt 8360

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291  ax-pre-lttrn 8293

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365

This theorem is used by:  xrltnsym2  10196  xrltle  10200