Theorem ixxdisj 9969

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

Hypotheses

Ref	Expression
ixxssixx.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, x   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( y, z)

Proof of Theorem ixxdisj

Step	Hyp	Ref	Expression
1		elin 3342	. . . 4 |- ( w e. ( ( A O B ) i^i ( B P C ) ) <-> ( w e. ( A O B ) /\ w e. ( B P C ) ) )
2		ixxssixx.1	. . . . . . . . . . 11 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
3	2	elixx1 9963	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
4	3	3adant3 1019	. . . . . . . . 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 296	. . . . . . . 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 1013	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( A O B ) ) -> w S B )
7	6	adantrr 479	. . . . . 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 9963	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C e. RR* ) -> ( w e. ( B P C ) <-> ( w e. RR* /\ B T w /\ w U C ) ) )
10	9	3adant1 1017	. . . . . . . . . 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 296	. . . . . . . . 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 1012	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> B T w )
13		simpl2 1003	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> B e. RR* )
14	11	simp1d 1011	. . . . . . . . 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 411	. . . . . . . 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 147	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> -. w S B )
18	17	adantrl 478	. . . . . 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 662	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. ( w e. ( A O B ) /\ w e. ( B P C ) ) )
20	19	pm2.21d 620	. . . 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	biimtrid 152	. . 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 3185	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A O B ) i^i ( B P C ) ) C_ (/) )
23		ss0 3487	. 2 |- ( ( ( A O B ) i^i ( B P C ) ) C_ (/) -> ( ( A O B ) i^i ( B P C ) ) = (/) )
24	22, 23	syl 14	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A O B ) i^i ( B P C ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   {crab 2476   i^i cin 3152   C_ wss 3153   (/)c0 3446   class class class wbr 4029  (class class class)co 5918   e. cmpo 5920   RR*cxr 8053

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058

This theorem is referenced by:  ioodisj  10059