Theorem ixxssixx 9678

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	elmpocl 5961	. . 3 |- ( w e. ( A O B ) -> ( A e. RR* /\ B e. RR* ) )
3		simp1 981	. . . . . 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 108	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> A e. RR* )
6		3simpa 978	. . . . . 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 360	. . . . . 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 109	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
11		3simpb 979	. . . . . 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 266	. . . . . . 7 |- ( ( B e. RR* /\ w e. RR* ) -> ( w S B -> w U B ) )
14	13	expimpd 360	. . . . . 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 1162	. . . 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 9673	. . . 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 9673	. . . 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 202	. . 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 3096	1 |- ( A O B ) C_ ( A P B )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   {crab 2418   C_ wss 3066   class class class wbr 3924  (class class class)co 5767   e. cmpo 5769   RR*cxr 7792

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797

This theorem is referenced by:  ioossicc  9735  icossicc  9736  iocssicc  9737  ioossico  9738