Theorem icodisj 10184

Description: End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)

Assertion

Ref	Expression
icodisj	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) = (/) )

Proof of Theorem icodisj

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3387	. . . 4 |- ( x e. ( ( A [,) B ) i^i ( B [,) C ) ) <-> ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) )
2		elico1 10115	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
3	2	3adant3 1041	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
4	3	biimpa 296	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( A [,) B ) ) -> ( x e. RR* /\ A <_ x /\ x < B ) )
5	4	simp3d 1035	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( A [,) B ) ) -> x < B )
6	5	adantrr 479	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) ) -> x < B )
7		elico1 10115	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
8	7	3adant1 1039	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
9	8	biimpa 296	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> ( x e. RR* /\ B <_ x /\ x < C ) )
10	9	simp2d 1034	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> B <_ x )
11		simpl2 1025	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> B e. RR* )
12	9	simp1d 1033	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> x e. RR* )
13		xrlenlt 8207	. . . . . . . . 9 |- ( ( B e. RR* /\ x e. RR* ) -> ( B <_ x <-> -. x < B ) )
14	11, 12, 13	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> ( B <_ x <-> -. x < B ) )
15	10, 14	mpbid 147	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> -. x < B )
16	15	adantrl 478	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) ) -> -. x < B )
17	6, 16	pm2.65da 665	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) )
18	17	pm2.21d 622	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) -> x e. (/) ) )
19	1, 18	biimtrid 152	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( ( A [,) B ) i^i ( B [,) C ) ) -> x e. (/) ) )
20	19	ssrdv 3230	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) C_ (/) )
21		ss0 3532	. 2 |- ( ( ( A [,) B ) i^i ( B [,) C ) ) C_ (/) -> ( ( A [,) B ) i^i ( B [,) C ) ) = (/) )
22	20, 21	syl 14	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   i^i cin 3196   C_ wss 3197   (/)c0 3491   class class class wbr 4082  (class class class)co 6000   RR*cxr 8176   < clt 8177   <_ cle 8178   [,)cico 10082

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-le 8183  df-ico 10086

This theorem is referenced by: (None)