Theorem iooneg 10267

Description: Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)

Assertion

Ref	Expression
iooneg	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A (,) B ) <-> -u C e. ( -u B (,) -u A ) ) )

Proof of Theorem iooneg

Step	Hyp	Ref	Expression
1		ltneg 8684	. . . . 5 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
2	1	3adant2 1043	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
3		ltneg 8684	. . . . . 6 |- ( ( C e. RR /\ B e. RR ) -> ( C < B <-> -u B < -u C ) )
4	3	ancoms 268	. . . . 5 |- ( ( B e. RR /\ C e. RR ) -> ( C < B <-> -u B < -u C ) )
5	4	3adant1 1042	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C < B <-> -u B < -u C ) )
6	2, 5	anbi12d 473	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < C /\ C < B ) <-> ( -u C < -u A /\ -u B < -u C ) ) )
7		ancom 266	. . 3 |- ( ( -u C < -u A /\ -u B < -u C ) <-> ( -u B < -u C /\ -u C < -u A ) )
8	6, 7	bitrdi 196	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < C /\ C < B ) <-> ( -u B < -u C /\ -u C < -u A ) ) )
9		rexr 8267	. . 3 |- ( A e. RR -> A e. RR* )
10		rexr 8267	. . 3 |- ( B e. RR -> B e. RR* )
11		rexr 8267	. . 3 |- ( C e. RR -> C e. RR* )
12		elioo5 10212	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) )
13	9, 10, 11, 12	syl3an 1316	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) )
14		renegcl 8482	. . . 4 |- ( B e. RR -> -u B e. RR )
15		renegcl 8482	. . . 4 |- ( A e. RR -> -u A e. RR )
16		renegcl 8482	. . . 4 |- ( C e. RR -> -u C e. RR )
17		rexr 8267	. . . . 5 |- ( -u B e. RR -> -u B e. RR* )
18		rexr 8267	. . . . 5 |- ( -u A e. RR -> -u A e. RR* )
19		rexr 8267	. . . . 5 |- ( -u C e. RR -> -u C e. RR* )
20		elioo5 10212	. . . . 5 |- ( ( -u B e. RR* /\ -u A e. RR* /\ -u C e. RR* ) -> ( -u C e. ( -u B (,) -u A ) <-> ( -u B < -u C /\ -u C < -u A ) ) )
21	17, 18, 19, 20	syl3an 1316	. . . 4 |- ( ( -u B e. RR /\ -u A e. RR /\ -u C e. RR ) -> ( -u C e. ( -u B (,) -u A ) <-> ( -u B < -u C /\ -u C < -u A ) ) )
22	14, 15, 16, 21	syl3an 1316	. . 3 |- ( ( B e. RR /\ A e. RR /\ C e. RR ) -> ( -u C e. ( -u B (,) -u A ) <-> ( -u B < -u C /\ -u C < -u A ) ) )
23	22	3com12 1234	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -u C e. ( -u B (,) -u A ) <-> ( -u B < -u C /\ -u C < -u A ) ) )
24	8, 13, 23	3bitr4d 220	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A (,) B ) <-> -u C e. ( -u B (,) -u A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   RRcr 8074   RR*cxr 8255   < clt 8256   -ucneg 8393   (,)cioo 10167

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-sub 8394  df-neg 8395  df-ioo 10171

This theorem is referenced by: (None)