Theorem iooneg 10080

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 8506	. . . . 5 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
2	1	3adant2 1018	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
3		ltneg 8506	. . . . . 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 1017	. . . 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 8089	. . 3 |- ( A e. RR -> A e. RR* )
10		rexr 8089	. . 3 |- ( B e. RR -> B e. RR* )
11		rexr 8089	. . 3 |- ( C e. RR -> C e. RR* )
12		elioo5 10025	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) )
13	9, 10, 11, 12	syl3an 1291	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) )
14		renegcl 8304	. . . 4 |- ( B e. RR -> -u B e. RR )
15		renegcl 8304	. . . 4 |- ( A e. RR -> -u A e. RR )
16		renegcl 8304	. . . 4 |- ( C e. RR -> -u C e. RR )
17		rexr 8089	. . . . 5 |- ( -u B e. RR -> -u B e. RR* )
18		rexr 8089	. . . . 5 |- ( -u A e. RR -> -u A e. RR* )
19		rexr 8089	. . . . 5 |- ( -u C e. RR -> -u C e. RR* )
20		elioo5 10025	. . . . 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 1291	. . . 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 1291	. . 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 1209	. 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 980   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   RRcr 7895   RR*cxr 8077   < clt 8078   -ucneg 8215   (,)cioo 9980

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-sub 8216  df-neg 8217  df-ioo 9984

This theorem is referenced by: (None)