Theorem ixxdisj 10137

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 3390	. . . 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 10131	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
4	3	3adant3 1043	. . . . . . . . 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 1037	. . . . . . 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 10131	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C e. RR* ) -> ( w e. ( B P C ) <-> ( w e. RR* /\ B T w /\ w U C ) ) )
10	9	3adant1 1041	. . . . . . . . . 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 1036	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> B T w )
13		simpl2 1027	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ w e. ( B P C ) ) -> B e. RR* )
14	11	simp1d 1035	. . . . . . . . 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 667	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. ( w e. ( A O B ) /\ w e. ( B P C ) ) )
20	19	pm2.21d 624	. . . 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 3233	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A O B ) i^i ( B P C ) ) C_ (/) )
23		ss0 3535	. 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 1004   = wceq 1397   e. wcel 2202   {crab 2514   i^i cin 3199   C_ wss 3200   (/)c0 3494   class class class wbr 4088  (class class class)co 6017   e. cmpo 6019   RR*cxr 8212

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217

This theorem is referenced by:  ioodisj  10227