Theorem ixxssixx 8771

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 5698	. . 3 |- ( w e. ( A O B ) -> ( A e. RR* /\ B e. RR* ) )
3		simp1 904	. . . . . 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 102	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> A e. RR* )
6		3simpa 901	. . . . . 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 345	. . . . . 6 |- ( A e. RR* -> ( ( w e. RR* /\ A R w ) -> A T w ) )
9	5, 6, 8	syl2im 34	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( w e. RR* /\ A R w /\ w S B ) -> A T w ) )
10		simpr 103	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
11		3simpb 902	. . . . . 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 255	. . . . . . 7 |- ( ( B e. RR* /\ w e. RR* ) -> ( w S B -> w U B ) )
14	13	expimpd 345	. . . . . 6 |- ( B e. RR* -> ( ( w e. RR* /\ w S B ) -> w U B ) )
15	10, 11, 14	syl2im 34	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( w e. RR* /\ A R w /\ w S B ) -> w U B ) )
16	4, 9, 15	3jcad 1085	. . . 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 8766	. . . 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 8766	. . . 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 192	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) -> w e. ( A P B ) ) )
21	2, 20	mpcom 32	. 2 |- ( w e. ( A O B ) -> w e. ( A P B ) )
22	21	ssriv 2949	1 |- ( A O B ) C_ ( A P B )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393   {crab 2310   C_ wss 2917   class class class wbr 3764  (class class class)co 5512   |-> cmpt2 5514   RR*cxr 7059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064

This theorem is referenced by:  ioossicc  8828  icossicc  8829  iocssicc  8830  ioossico  8831