Theorem ixxssixx 8925

Description: An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Hypotheses

Ref	Expression
ixxssixx.1	|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
ixx.2	|- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x T z /\ z U y ) } )
ixx.3	|- ( ( A e. RR* /\ w e. RR* ) -> ( A R w -> A T w ) )
ixx.4	|- ( ( w e. RR* /\ B e. RR* ) -> ( w S B -> w U B ) )

Assertion

Ref	Expression
ixxssixx	|- ( A O B ) C_ ( A P B )

Distinct variable groups:   x, w, y, z, A   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 ixxssixx

Step	Hyp	Ref	Expression
1		ixxssixx.1	. . . 4 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
2	1	elmpt2cl 5718	. . 3 |- ( w e. ( A O B ) -> ( A e. RR* /\ B e. RR* ) )
3		simp1 938	. . . . . 6 |- ( ( w e. RR* /\ A R w /\ w S B ) -> w e. RR* )
4	3	a1i 9	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( w e. RR* /\ A R w /\ w S B ) -> w e. RR* ) )
5		simpl 107	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> A e. RR* )
6		3simpa 935	. . . . . 6 |- ( ( w e. RR* /\ A R w /\ w S B ) -> ( w e. RR* /\ A R w ) )
7		ixx.3	. . . . . . 7 |- ( ( A e. RR* /\ w e. RR* ) -> ( A R w -> A T w ) )
8	7	expimpd 355	. . . . . 6 |- ( A e. RR* -> ( ( w e. RR* /\ A R w ) -> A T w ) )
9	5, 6, 8	syl2im 38	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( w e. RR* /\ A R w /\ w S B ) -> A T w ) )
10		simpr 108	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
11		3simpb 936	. . . . . 6 |- ( ( w e. RR* /\ A R w /\ w S B ) -> ( w e. RR* /\ w S B ) )
12		ixx.4	. . . . . . . 8 |- ( ( w e. RR* /\ B e. RR* ) -> ( w S B -> w U B ) )
13	12	ancoms 264	. . . . . . 7 |- ( ( B e. RR* /\ w e. RR* ) -> ( w S B -> w U B ) )
14	13	expimpd 355	. . . . . 6 |- ( B e. RR* -> ( ( w e. RR* /\ w S B ) -> w U B ) )
15	10, 11, 14	syl2im 38	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( w e. RR* /\ A R w /\ w S B ) -> w U B ) )
16	4, 9, 15	3jcad 1119	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( w e. RR* /\ A R w /\ w S B ) -> ( w e. RR* /\ A T w /\ w U B ) ) )
17	1	elixx1 8920	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
18		ixx.2	. . . . 5 |- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x T z /\ z U y ) } )
19	18	elixx1 8920	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A P B ) <-> ( w e. RR* /\ A T w /\ w U B ) ) )
20	16, 17, 19	3imtr4d 201	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) -> w e. ( A P B ) ) )
21	2, 20	mpcom 36	. 2 |- ( w e. ( A O B ) -> w e. ( A P B ) )
22	21	ssriv 3003	1 |- ( A O B ) C_ ( A P B )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   {crab 2352   C_ wss 2973   class class class wbr 3785  (class class class)co 5532   |-> cmpt2 5534   RR*cxr 7152

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157

This theorem is referenced by:  ioossicc  8982  icossicc  8983  iocssicc  8984  ioossico  8985