Theorem ixxdisj 12190

Description: Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)

Hypotheses

Ref	Expression
ixx.1	|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
ixxun.2	|- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x T z /\ z U y ) } )
ixxun.3	|- ( ( B e. RR* /\ w e. RR* ) -> ( B T w <-> -. w S B ) )

Assertion

Ref	Expression
ixxdisj	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A O B ) i^i ( B P C ) ) = (/) )

Distinct variable groups:   x, w, y, z, A   w, C, x, y, z   w, O   w, B, x, y, z   w, P   x, R, y, z   x, S, y, z   x, T, y, z   x, U, y, z

Allowed substitution hints:   P( x, y, z)   R( w)   S( w)   T( w)   U( w)   O( x, y, z)

Proof of Theorem ixxdisj

Step	Hyp	Ref	Expression
1		elin 3796	. . . 4 |- ( w e. ( ( A O B ) i^i ( B P C ) ) <-> ( w e. ( A O B ) /\ w e. ( B P C ) ) )
2		ixx.1	. . . . . . . . . . 11 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
3	2	elixx1 12184	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
4	3	3adant3 1081	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
5	4	biimpa 501	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( A O B ) ) -> ( w e. RR* /\ A R w /\ w S B ) )
6	5	simp3d 1075	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( A O B ) ) -> w S B )
7	6	adantrr 753	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( w e. ( A O B ) /\ w e. ( B P C ) ) ) -> w S B )
8		ixxun.2	. . . . . . . . . . . 12 |- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x T z /\ z U y ) } )
9	8	elixx1 12184	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C e. RR* ) -> ( w e. ( B P C ) <-> ( w e. RR* /\ B T w /\ w U C ) ) )
10	9	3adant1 1079	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( w e. ( B P C ) <-> ( w e. RR* /\ B T w /\ w U C ) ) )
11	10	biimpa 501	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> ( w e. RR* /\ B T w /\ w U C ) )
12	11	simp2d 1074	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> B T w )
13		simpl2 1065	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> B e. RR* )
14	11	simp1d 1073	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> w e. RR* )
15		ixxun.3	. . . . . . . . 9 |- ( ( B e. RR* /\ w e. RR* ) -> ( B T w <-> -. w S B ) )
16	13, 14, 15	syl2anc 693	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> ( B T w <-> -. w S B ) )
17	12, 16	mpbid 222	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> -. w S B )
18	17	adantrl 752	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( w e. ( A O B ) /\ w e. ( B P C ) ) ) -> -. w S B )
19	7, 18	pm2.65da 600	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. ( w e. ( A O B ) /\ w e. ( B P C ) ) )
20	19	pm2.21d 118	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( w e. ( A O B ) /\ w e. ( B P C ) ) -> w e. (/) ) )
21	1, 20	syl5bi 232	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( w e. ( ( A O B ) i^i ( B P C ) ) -> w e. (/) ) )
22	21	ssrdv 3609	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A O B ) i^i ( B P C ) ) C_ (/) )
23		ss0 3974	. 2 |- ( ( ( A O B ) i^i ( B P C ) ) C_ (/) -> ( ( A O B ) i^i ( B P C ) ) = (/) )
24	22, 23	syl 17	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A O B ) i^i ( B P C ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   |-> cmpt2 6652   RR*cxr 10073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-xr 10078

This theorem is referenced by:  ioodisj  12302  lecldbas  21023  icopnfcld  22571  iocmnfcld  22572  ioombl  23333  ismbf3d  23421  joiniooico  29536  asindmre  33495  dvasin  33496