Theorem ixxssixx 10024

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 6141	. . 3 |- ( w e. ( A O B ) -> ( A e. RR* /\ B e. RR* ) )
3		simp1 1000	. . . . . 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 109	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> A e. RR* )
6		3simpa 997	. . . . . 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 363	. . . . . 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 110	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
11		3simpb 998	. . . . . 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 268	. . . . . . 7 |- ( ( B e. RR* /\ w e. RR* ) -> ( w S B -> w U B ) )
14	13	expimpd 363	. . . . . 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 1181	. . . 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 10019	. . . 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 10019	. . . 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 203	. . 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 3197	1 |- ( A O B ) C_ ( A P B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176   {crab 2488   C_ wss 3166   class class class wbr 4044  (class class class)co 5944   e. cmpo 5946   RR*cxr 8106

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111

This theorem is referenced by:  ioossicc  10081  icossicc  10082  iocssicc  10083  ioossico  10084