Theorem xrltnsym 6725

Description: Ordering on the extended reals is not symmetric.

Assertion

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

Proof of Theorem xrltnsym

Step	Hyp	Ref	Expression
1		ltnsym 6702	. . . 4 |- ((A e. RR /\ B e. RR) -> (A < B -> -. B < A))
2		rexr 6668	. . . . . . . 8 |- (A e. RR -> A e. RR*)
3		pnfnlt 6721	. . . . . . . 8 |- (A e. RR* -> -. +oo < A)
4	2, 3	syl 12	. . . . . . 7 |- (A e. RR -> -. +oo < A)
5	4	adantr 425	. . . . . 6 |- ((A e. RR /\ B = +oo) -> -. +oo < A)
6		breq1 3341	. . . . . . 7 |- (B = +oo -> (B < A <-> +oo < A))
7	6	adantl 424	. . . . . 6 |- ((A e. RR /\ B = +oo) -> (B < A <-> +oo < A))
8	5, 7	mtbird 783	. . . . 5 |- ((A e. RR /\ B = +oo) -> -. B < A)
9	8	a1d 15	. . . 4 |- ((A e. RR /\ B = +oo) -> (A < B -> -. B < A))
10		nltmnf 6722	. . . . . . . 8 |- (A e. RR* -> -. A < -oo)
11	2, 10	syl 12	. . . . . . 7 |- (A e. RR -> -. A < -oo)
12	11	adantr 425	. . . . . 6 |- ((A e. RR /\ B = -oo) -> -. A < -oo)
13		breq2 3342	. . . . . . 7 |- (B = -oo -> (A < B <-> A < -oo))
14	13	adantl 424	. . . . . 6 |- ((A e. RR /\ B = -oo) -> (A < B <-> A < -oo))
15	12, 14	mtbird 783	. . . . 5 |- ((A e. RR /\ B = -oo) -> -. A < B)
16	15	pm2.21d 94	. . . 4 |- ((A e. RR /\ B = -oo) -> (A < B -> -. B < A))
17	1, 9, 16	3jaodan 1163	. . 3 |- ((A e. RR /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
18		pnfnlt 6721	. . . . . . 7 |- (B e. RR* -> -. +oo < B)
19	18	adantl 424	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> -. +oo < B)
20		breq1 3341	. . . . . . 7 |- (A = +oo -> (A < B <-> +oo < B))
21	20	adantr 425	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> (A < B <-> +oo < B))
22	19, 21	mtbird 783	. . . . 5 |- ((A = +oo /\ B e. RR*) -> -. A < B)
23	22	pm2.21d 94	. . . 4 |- ((A = +oo /\ B e. RR*) -> (A < B -> -. B < A))
24		elxr 6706	. . . 4 |- (B e. RR* <-> (B e. RR \/ B = +oo \/ B = -oo))
25	23, 24	sylan2br 502	. . 3 |- ((A = +oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
26		rexr 6668	. . . . . . . 8 |- (B e. RR -> B e. RR*)
27		nltmnf 6722	. . . . . . . 8 |- (B e. RR* -> -. B < -oo)
28	26, 27	syl 12	. . . . . . 7 |- (B e. RR -> -. B < -oo)
29	28	adantl 424	. . . . . 6 |- ((A = -oo /\ B e. RR) -> -. B < -oo)
30		breq2 3342	. . . . . . 7 |- (A = -oo -> (B < A <-> B < -oo))
31	30	adantr 425	. . . . . 6 |- ((A = -oo /\ B e. RR) -> (B < A <-> B < -oo))
32	29, 31	mtbird 783	. . . . 5 |- ((A = -oo /\ B e. RR) -> -. B < A)
33	32	a1d 15	. . . 4 |- ((A = -oo /\ B e. RR) -> (A < B -> -. B < A))
34		mnfxr 6662	. . . . . . . 8 |- -oo e. RR*
35		pnfnlt 6721	. . . . . . . 8 |- ( -oo e. RR* -> -. +oo < -oo)
36	34, 35	ax-mp 7	. . . . . . 7 |- -. +oo < -oo
37		breq12 3343	. . . . . . 7 |- ((B = +oo /\ A = -oo) -> (B < A <-> +oo < -oo))
38	36, 37	mtbiri 785	. . . . . 6 |- ((B = +oo /\ A = -oo) -> -. B < A)
39	38	ancoms 484	. . . . 5 |- ((A = -oo /\ B = +oo) -> -. B < A)
40	39	a1d 15	. . . 4 |- ((A = -oo /\ B = +oo) -> (A < B -> -. B < A))
41		xrltnr 6716	. . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo)
42	34, 41	ax-mp 7	. . . . . 6 |- -. -oo < -oo
43		breq12 3343	. . . . . 6 |- ((A = -oo /\ B = -oo) -> (A < B <-> -oo < -oo))
44	42, 43	mtbiri 785	. . . . 5 |- ((A = -oo /\ B = -oo) -> -. A < B)
45	44	pm2.21d 94	. . . 4 |- ((A = -oo /\ B = -oo) -> (A < B -> -. B < A))
46	33, 40, 45	3jaodan 1163	. . 3 |- ((A = -oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
47	17, 25, 46	3jaoian 1162	. 2 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> -. B < A))
48		elxr 6706	. 2 |- (A e. RR* <-> (A e. RR \/ A = +oo \/ A = -oo))
49	47, 48, 24	syl2anb 504	1 |- ((A e. RR* /\ B e. RR*) -> (A < B -> -. B < A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300   class class class wbr 3338  RRcr 6385   +oocpnf 6650   -oocmnf 6651  RR*cxr 6652   < clt 6653

This theorem is referenced by:  xrltnsym2 6726  xrlttri 6727

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-ltp 6242  df-enr 6318  df-nr 6319  df-ltr 6322  df-0r 6323  df-c 6392  df-r 6396  df-lt 6399  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657

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