Theorem iooneg 10110

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 8535	. . . . 5 |- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
2	1	3adant2 1019	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> -u C < -u A ) )
3		ltneg 8535	. . . . . 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 1018	. . . 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 8118	. . 3 |- ( A e. RR -> A e. RR* )
10		rexr 8118	. . 3 |- ( B e. RR -> B e. RR* )
11		rexr 8118	. . 3 |- ( C e. RR -> C e. RR* )
12		elioo5 10055	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) )
13	9, 10, 11, 12	syl3an 1292	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A (,) B ) <-> ( A < C /\ C < B ) ) )
14		renegcl 8333	. . . 4 |- ( B e. RR -> -u B e. RR )
15		renegcl 8333	. . . 4 |- ( A e. RR -> -u A e. RR )
16		renegcl 8333	. . . 4 |- ( C e. RR -> -u C e. RR )
17		rexr 8118	. . . . 5 |- ( -u B e. RR -> -u B e. RR* )
18		rexr 8118	. . . . 5 |- ( -u A e. RR -> -u A e. RR* )
19		rexr 8118	. . . . 5 |- ( -u C e. RR -> -u C e. RR* )
20		elioo5 10055	. . . . 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 1292	. . . 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 1292	. . 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 1210	. 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 981   e. wcel 2176   class class class wbr 4044  (class class class)co 5944   RRcr 7924   RR*cxr 8106   < clt 8107   -ucneg 8244   (,)cioo 10010

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-sub 8245  df-neg 8246  df-ioo 10014

This theorem is referenced by: (None)