Theorem ixxss2 9980

Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Hypotheses

Ref	Expression
ixxssixx.1	|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
ixxss2.2	|- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z T y ) } )
ixxss2.3	|- ( ( w e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( w T B /\ B W C ) -> w S C ) )

Assertion

Ref	Expression
ixxss2	|- ( ( C e. RR* /\ B W C ) -> ( A P B ) C_ ( A O C ) )

Distinct variable groups:   x, w, y, z, A   w, C, x, y, z   w, O, x   w, B, x, y, z   w, P   x, R, y, z   x, S, y, z   x, T, y, z   w, W

Allowed substitution hints:   P( x, y, z)   R( w)   S( w)   T( w)   O( y, z)   W( x, y, z)

Proof of Theorem ixxss2

Step	Hyp	Ref	Expression
1		ixxss2.2	. . . . . . . 8 |- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z T y ) } )
2	1	elixx3g 9976	. . . . . . 7 |- ( w e. ( A P B ) <-> ( ( A e. RR* /\ B e. RR* /\ w e. RR* ) /\ ( A R w /\ w T B ) ) )
3	2	simplbi 274	. . . . . 6 |- ( w e. ( A P B ) -> ( A e. RR* /\ B e. RR* /\ w e. RR* ) )
4	3	adantl 277	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> ( A e. RR* /\ B e. RR* /\ w e. RR* ) )
5	4	simp3d 1013	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w e. RR* )
6	2	simprbi 275	. . . . . 6 |- ( w e. ( A P B ) -> ( A R w /\ w T B ) )
7	6	adantl 277	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> ( A R w /\ w T B ) )
8	7	simpld 112	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> A R w )
9	7	simprd 114	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w T B )
10		simplr 528	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> B W C )
11	4	simp2d 1012	. . . . . 6 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> B e. RR* )
12		simpll 527	. . . . . 6 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> C e. RR* )
13		ixxss2.3	. . . . . 6 |- ( ( w e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( w T B /\ B W C ) -> w S C ) )
14	5, 11, 12, 13	syl3anc 1249	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> ( ( w T B /\ B W C ) -> w S C ) )
15	9, 10, 14	mp2and 433	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w S C )
16	4	simp1d 1011	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> A e. RR* )
17		ixxssixx.1	. . . . . 6 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
18	17	elixx1 9972	. . . . 5 |- ( ( A e. RR* /\ C e. RR* ) -> ( w e. ( A O C ) <-> ( w e. RR* /\ A R w /\ w S C ) ) )
19	16, 12, 18	syl2anc 411	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> ( w e. ( A O C ) <-> ( w e. RR* /\ A R w /\ w S C ) ) )
20	5, 8, 15, 19	mpbir3and 1182	. . 3 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w e. ( A O C ) )
21	20	ex 115	. 2 |- ( ( C e. RR* /\ B W C ) -> ( w e. ( A P B ) -> w e. ( A O C ) ) )
22	21	ssrdv 3189	1 |- ( ( C e. RR* /\ B W C ) -> ( A P B ) C_ ( A O C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   {crab 2479   C_ wss 3157   class class class wbr 4033  (class class class)co 5922   e. cmpo 5924   RR*cxr 8060

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065

This theorem is referenced by:  iooss2  9992