Theorem ixxdisj 9978

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 3346	. . . 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 9972	. . . . . . . . . 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 9972	. . . . . . . . . . 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 3189	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A O B ) i^i ( B P C ) ) C_ (/) )
23		ss0 3491	. 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 2167   {crab 2479   i^i cin 3156   C_ wss 3157   (/)c0 3450   class class class wbr 4033  (class class class)co 5922   e. cmpo 5924   RR*cxr 8060

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971

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-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065

This theorem is referenced by:  ioodisj  10068