Theorem xrltnsymt 5550

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 5532	. . . 4 |- ((A e. RR /\ B e. RR) -> (A < B -> -. B < A))
2		rexrt 5499	. . . . . . . 8 |- (A e. RR -> A e. RR*)
3		pnfnltt 5546	. . . . . . . 8 |- (A e. RR* -> -. +oo < A)
4	2, 3	syl 10	. . . . . . 7 |- (A e. RR -> -. +oo < A)
5	4	adantr 389	. . . . . 6 |- ((A e. RR /\ B = +oo) -> -. +oo < A)
6		breq1 2622	. . . . . . 7 |- (B = +oo -> (B < A <-> +oo < A))
7	6	adantl 388	. . . . . 6 |- ((A e. RR /\ B = +oo) -> (B < A <-> +oo < A))
8	5, 7	mtbird 715	. . . . 5 |- ((A e. RR /\ B = +oo) -> -. B < A)
9	8	a1d 12	. . . 4 |- ((A e. RR /\ B = +oo) -> (A < B -> -. B < A))
10		nltmnft 5547	. . . . . . . 8 |- (A e. RR* -> -. A < -oo)
11	2, 10	syl 10	. . . . . . 7 |- (A e. RR -> -. A < -oo)
12	11	adantr 389	. . . . . 6 |- ((A e. RR /\ B = -oo) -> -. A < -oo)
13		breq2 2623	. . . . . . 7 |- (B = -oo -> (A < B <-> A < -oo))
14	13	adantl 388	. . . . . 6 |- ((A e. RR /\ B = -oo) -> (A < B <-> A < -oo))
15	12, 14	mtbird 715	. . . . 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 890	. . 3 |- ((A e. RR /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
18		pnfnltt 5546	. . . . . . 7 |- (B e. RR* -> -. +oo < B)
19	18	adantl 388	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> -. +oo < B)
20		breq1 2622	. . . . . . 7 |- (A = +oo -> (A < B <-> +oo < B))
21	20	adantr 389	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> (A < B <-> +oo < B))
22	19, 21	mtbird 715	. . . . 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 5535	. . . 4 |- (B e. RR* <-> (B e. RR \/ B = +oo \/ B = -oo))
25	23, 24	sylan2br 453	. . 3 |- ((A = +oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
26		rexrt 5499	. . . . . . . 8 |- (B e. RR -> B e. RR*)
27		nltmnft 5547	. . . . . . . 8 |- (B e. RR* -> -. B < -oo)
28	26, 27	syl 10	. . . . . . 7 |- (B e. RR -> -. B < -oo)
29	28	adantl 388	. . . . . 6 |- ((A = -oo /\ B e. RR) -> -. B < -oo)
30		breq2 2623	. . . . . . 7 |- (A = -oo -> (B < A <-> B < -oo))
31	30	adantr 389	. . . . . 6 |- ((A = -oo /\ B e. RR) -> (B < A <-> B < -oo))
32	29, 31	mtbird 715	. . . . 5 |- ((A = -oo /\ B e. RR) -> -. B < A)
33	32	a1d 12	. . . 4 |- ((A = -oo /\ B e. RR) -> (A < B -> -. B < A))
34		mnfxr 5494	. . . . . . . 8 |- -oo e. RR*
35		pnfnltt 5546	. . . . . . . 8 |- ( -oo e. RR* -> -. +oo < -oo)
36	34, 35	ax-mp 7	. . . . . . 7 |- -. +oo < -oo
37		breq12 2624	. . . . . . 7 |- ((B = +oo /\ A = -oo) -> (B < A <-> +oo < -oo))
38	36, 37	mtbiri 717	. . . . . 6 |- ((B = +oo /\ A = -oo) -> -. B < A)
39	38	ancoms 436	. . . . 5 |- ((A = -oo /\ B = +oo) -> -. B < A)
40	39	a1d 12	. . . 4 |- ((A = -oo /\ B = +oo) -> (A < B -> -. B < A))
41		xrltnrt 5541	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo)
42	34, 41	ax-mp 7	. . . . . 6 |- -. -oo < -oo
43		breq12 2624	. . . . . 6 |- ((A = -oo /\ B = -oo) -> (A < B <-> -oo < -oo))
44	42, 43	mtbiri 717	. . . . 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 890	. . 3 |- ((A = -oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
47	17, 25, 46	3jaoian 889	. 2 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
48		elxr 5535	. 2 |- (A e. RR* <-> (A e. RR \/ A = +oo \/ A = -oo))
49	47, 48, 24	syl2anb 455	1 |- ((A e. RR* /\ B e. RR*) -> (A < B -> -. B < A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   \/ w3o 774   = wceq 956   e. wcel 958   class class class wbr 2619  RRcr 5233   +oocpnf 5483   -oocmnf 5484  RR*cxr 5485   < clt 5486

This theorem is referenced by:  xrltnsym2t 5551  xrlttrit 5552

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-ltp 5090  df-enr 5166  df-nr 5167  df-ltr 5170  df-0r 5171  df-c 5240  df-r 5244  df-lt 5247  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490

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