Theorem ixxssixx 10136

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 6216	. . 3 |- ( w e. ( A O B ) -> ( A e. RR* /\ B e. RR* ) )
3		simp1 1023	. . . . . 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 1020	. . . . . 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 1021	. . . . . 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 1204	. . . 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 10131	. . . 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 10131	. . . 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 3231	1 |- ( A O B ) C_ ( A P B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   {crab 2514   C_ wss 3200   class class class wbr 4088  (class class class)co 6017   e. cmpo 6019   RR*cxr 8212

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217

This theorem is referenced by:  ioossicc  10193  icossicc  10194  iocssicc  10195  ioossico  10196