Theorem iooneg 10369

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 8780	. . . . 5 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
2	1	3adant2 1047	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
3		ltneg 8780	. . . . . 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 1046	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C < B <-> -u B < -u C ) )
6	2, 5	anbi12d 477	. . 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 8361	. . 3 |- ( A e. RR -> A e. RR* )
10		rexr 8361	. . 3 |- ( B e. RR -> B e. RR* )
11		rexr 8361	. . 3 |- ( C e. RR -> C e. RR* )
12		elioo5 10314	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) )
13	9, 10, 11, 12	syl3an 1320	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) )
14		renegcl 8577	. . . 4 |- ( B e. RR -> -u B e. RR )
15		renegcl 8577	. . . 4 |- ( A e. RR -> -u A e. RR )
16		renegcl 8577	. . . 4 |- ( C e. RR -> -u C e. RR )
17		rexr 8361	. . . . 5 |- ( -u B e. RR -> -u B e. RR* )
18		rexr 8361	. . . . 5 |- ( -u A e. RR -> -u A e. RR* )
19		rexr 8361	. . . . 5 |- ( -u C e. RR -> -u C e. RR* )
20		elioo5 10314	. . . . 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 1320	. . . 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 1320	. . 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 1238	. 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 1009   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   RR*cxr 8349   < clt 8350   -ucneg 8488   (,)cioo 10269

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-sub 8489  df-neg 8490  df-ioo 10273

This theorem is referenced by: (None)