Theorem iooneg 9990

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 8421	. . . . 5 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
2	1	3adant2 1016	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
3		ltneg 8421	. . . . . 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 1015	. . . 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 8005	. . 3 |- ( A e. RR -> A e. RR* )
10		rexr 8005	. . 3 |- ( B e. RR -> B e. RR* )
11		rexr 8005	. . 3 |- ( C e. RR -> C e. RR* )
12		elioo5 9935	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) )
13	9, 10, 11, 12	syl3an 1280	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) )
14		renegcl 8220	. . . 4 |- ( B e. RR -> -u B e. RR )
15		renegcl 8220	. . . 4 |- ( A e. RR -> -u A e. RR )
16		renegcl 8220	. . . 4 |- ( C e. RR -> -u C e. RR )
17		rexr 8005	. . . . 5 |- ( -u B e. RR -> -u B e. RR* )
18		rexr 8005	. . . . 5 |- ( -u A e. RR -> -u A e. RR* )
19		rexr 8005	. . . . 5 |- ( -u C e. RR -> -u C e. RR* )
20		elioo5 9935	. . . . 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 1280	. . . 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 1280	. . 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 1207	. 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 978   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   RRcr 7812   RR*cxr 7993   < clt 7994   -ucneg 8131   (,)cioo 9890

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-sub 8132  df-neg 8133  df-ioo 9894

This theorem is referenced by: (None)