Theorem icodisj 9928

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 3305	. . . 4 |- ( x e. ( ( A [,) B ) i^i ( B [,) C ) ) <-> ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) )
2		elico1 9859	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
3	2	3adant3 1007	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
4	3	biimpa 294	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( A [,) B ) ) -> ( x e. RR* /\ A <_ x /\ x < B ) )
5	4	simp3d 1001	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( A [,) B ) ) -> x < B )
6	5	adantrr 471	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) ) -> x < B )
7		elico1 9859	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
8	7	3adant1 1005	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
9	8	biimpa 294	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> ( x e. RR* /\ B <_ x /\ x < C ) )
10	9	simp2d 1000	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> B <_ x )
11		simpl2 991	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> B e. RR* )
12	9	simp1d 999	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> x e. RR* )
13		xrlenlt 7963	. . . . . . . . 9 |- ( ( B e. RR* /\ x e. RR* ) -> ( B <_ x <-> -. x < B ) )
14	11, 12, 13	syl2anc 409	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> ( B <_ x <-> -. x < B ) )
15	10, 14	mpbid 146	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> -. x < B )
16	15	adantrl 470	. . . . . 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 651	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) )
18	17	pm2.21d 609	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) -> x e. (/) ) )
19	1, 18	syl5bi 151	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x e. ( ( A [,) B ) i^i ( B [,) C ) ) -> x e. (/) ) )
20	19	ssrdv 3148	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) C_ (/) )
21		ss0 3449	. 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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   i^i cin 3115   C_ wss 3116   (/)c0 3409   class class class wbr 3982  (class class class)co 5842   RR*cxr 7932   < clt 7933   <_ cle 7934   [,)cico 9826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-le 7939  df-ico 9830

This theorem is referenced by: (None)