Theorem xrltnsymt 5562

Description: Ordering on the extended reals is not symmetric.

Assertion

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

Proof of Theorem xrltnsymt

Step	Hyp	Ref	Expression
1		ltnsymt 5544	. . . 4 |- ((A e. RR /\ B e. RR) -> (A < B -> -. B < A))
2		rexrt 5511	. . . . . . . 8 |- (A e. RR -> A e. RR*)
3		pnfnltt 5558	. . . . . . . 8 |- (A e. RR* -> -. +oo < A)
4	2, 3	syl 10	. . . . . . 7 |- (A e. RR -> -. +oo < A)
5	4	adantr 391	. . . . . 6 |- ((A e. RR /\ B = +oo) -> -. +oo < A)
6		breq1 2627	. . . . . . 7 |- (B = +oo -> (B < A <-> +oo < A))
7	6	adantl 390	. . . . . 6 |- ((A e. RR /\ B = +oo) -> (B < A <-> +oo < A))
8	5, 7	mtbird 717	. . . . 5 |- ((A e. RR /\ B = +oo) -> -. B < A)
9	8	a1d 12	. . . 4 |- ((A e. RR /\ B = +oo) -> (A < B -> -. B < A))
10		nltmnft 5559	. . . . . . . 8 |- (A e. RR* -> -. A < -oo)
11	2, 10	syl 10	. . . . . . 7 |- (A e. RR -> -. A < -oo)
12	11	adantr 391	. . . . . 6 |- ((A e. RR /\ B = -oo) -> -. A < -oo)
13		breq2 2628	. . . . . . 7 |- (B = -oo -> (A < B <-> A < -oo))
14	13	adantl 390	. . . . . 6 |- ((A e. RR /\ B = -oo) -> (A < B <-> A < -oo))
15	12, 14	mtbird 717	. . . . 5 |- ((A e. RR /\ B = -oo) -> -. A < B)
16	15	pm2.21d 78	. . . 4 |- ((A e. RR /\ B = -oo) -> (A < B -> -. B < A))
17	1, 9, 16	3jaodan 892	. . 3 |- ((A e. RR /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
18		pnfnltt 5558	. . . . . . 7 |- (B e. RR* -> -. +oo < B)
19	18	adantl 390	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> -. +oo < B)
20		breq1 2627	. . . . . . 7 |- (A = +oo -> (A < B <-> +oo < B))
21	20	adantr 391	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> (A < B <-> +oo < B))
22	19, 21	mtbird 717	. . . . 5 |- ((A = +oo /\ B e. RR*) -> -. A < B)
23	22	pm2.21d 78	. . . 4 |- ((A = +oo /\ B e. RR*) -> (A < B -> -. B < A))
24		elxr 5547	. . . 4 |- (B e. RR* <-> (B e. RR \/ B = +oo \/ B = -oo))
25	23, 24	sylan2br 455	. . 3 |- ((A = +oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
26		rexrt 5511	. . . . . . . 8 |- (B e. RR -> B e. RR*)
27		nltmnft 5559	. . . . . . . 8 |- (B e. RR* -> -. B < -oo)
28	26, 27	syl 10	. . . . . . 7 |- (B e. RR -> -. B < -oo)
29	28	adantl 390	. . . . . 6 |- ((A = -oo /\ B e. RR) -> -. B < -oo)
30		breq2 2628	. . . . . . 7 |- (A = -oo -> (B < A <-> B < -oo))
31	30	adantr 391	. . . . . 6 |- ((A = -oo /\ B e. RR) -> (B < A <-> B < -oo))
32	29, 31	mtbird 717	. . . . 5 |- ((A = -oo /\ B e. RR) -> -. B < A)
33	32	a1d 12	. . . 4 |- ((A = -oo /\ B e. RR) -> (A < B -> -. B < A))
34		mnfxr 5506	. . . . . . . 8 |- -oo e. RR*
35		pnfnltt 5558	. . . . . . . 8 |- ( -oo e. RR* -> -. +oo < -oo)
36	34, 35	ax-mp 7	. . . . . . 7 |- -. +oo < -oo
37		breq12 2629	. . . . . . 7 |- ((B = +oo /\ A = -oo) -> (B < A <-> +oo < -oo))
38	36, 37	mtbiri 719	. . . . . 6 |- ((B = +oo /\ A = -oo) -> -. B < A)
39	38	ancoms 438	. . . . 5 |- ((A = -oo /\ B = +oo) -> -. B < A)
40	39	a1d 12	. . . 4 |- ((A = -oo /\ B = +oo) -> (A < B -> -. B < A))
41		xrltnrt 5553	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo)
42	34, 41	ax-mp 7	. . . . . 6 |- -. -oo < -oo
43		breq12 2629	. . . . . 6 |- ((A = -oo /\ B = -oo) -> (A < B <-> -oo < -oo))
44	42, 43	mtbiri 719	. . . . 5 |- ((A = -oo /\ B = -oo) -> -. A < B)
45	44	pm2.21d 78	. . . 4 |- ((A = -oo /\ B = -oo) -> (A < B -> -. B < A))
46	33, 40, 45	3jaodan 892	. . 3 |- ((A = -oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
47	17, 25, 46	3jaoian 891	. 2 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
48		elxr 5547	. 2 |- (A e. RR* <-> (A e. RR \/ A = +oo \/ A = -oo))
49	47, 48, 24	syl2anb 457	1 |- ((A e. RR* /\ B e. RR*) -> (A < B -> -. B < A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   \/ w3o 776   = wceq 958   e. wcel 960   class class class wbr 2624  RRcr 5245   +oocpnf 5495   -oocmnf 5496  RR*cxr 5497   < clt 5498

This theorem is referenced by:  xrltnsym2t 5563  xrlttrit 5564

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-ltp 5102  df-enr 5178  df-nr 5179  df-ltr 5182  df-0r 5183  df-c 5252  df-r 5256  df-lt 5259  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502

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