Theorem icodisj 10344

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 3406	. . . 4 |- ( x e. ( ( A [,) B ) i^i ( B [,) C ) ) <-> ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) )
2		elico1 10275	. . . . . . . . . 10 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
3	2	3adant3 1044	. . . . . . . . 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 1038	. . . . . . 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 10275	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C e. RR* ) -> ( x e. ( B [,) C ) <-> ( x e. RR* /\ B <_ x /\ x < C ) ) )
8	7	3adant1 1042	. . . . . . . . . 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 1037	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> B <_ x )
11		simpl2 1028	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> B e. RR* )
12	9	simp1d 1036	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ x e. ( B [,) C ) ) -> x e. RR* )
13		xrlenlt 8354	. . . . . . . . 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 667	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -. ( x e. ( A [,) B ) /\ x e. ( B [,) C ) ) )
18	17	pm2.21d 624	. . . 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 3248	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) C_ (/) )
21		ss0 3553	. 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 1005   = wceq 1398   e. wcel 2205   i^i cin 3213   C_ wss 3214   (/)c0 3512   class class class wbr 4114  (class class class)co 6058   RR*cxr 8323   < clt 8324   <_ cle 8325   [,)cico 10242

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-le 8330  df-ico 10246

This theorem is referenced by: (None)