Theorem ixxss1 10306

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 ) } )
ixxss1.2	|- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x T z /\ z S y ) } )
ixxss1.3	|- ( ( A e. RR* /\ B e. RR* /\ w e. RR* ) -> ( ( A W B /\ B T w ) -> A R w ) )

Assertion

Ref	Expression
ixxss1	|- ( ( A e. RR* /\ A W B ) -> ( B P C ) 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 ixxss1

Step	Hyp	Ref	Expression
1		ixxss1.2	. . . . . . . 8 |- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x T z /\ z S y ) } )
2	1	elixx3g 10303	. . . . . . 7 |- ( w e. ( B P C ) <-> ( ( B e. RR* /\ C e. RR* /\ w e. RR* ) /\ ( B T w /\ w S C ) ) )
3	2	simplbi 274	. . . . . 6 |- ( w e. ( B P C ) -> ( B e. RR* /\ C e. RR* /\ w e. RR* ) )
4	3	adantl 277	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> ( B e. RR* /\ C e. RR* /\ w e. RR* ) )
5	4	simp3d 1042	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> w e. RR* )
6		simplr 533	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> A W B )
7	2	simprbi 275	. . . . . . 7 |- ( w e. ( B P C ) -> ( B T w /\ w S C ) )
8	7	adantl 277	. . . . . 6 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> ( B T w /\ w S C ) )
9	8	simpld 112	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> B T w )
10		simpll 531	. . . . . 6 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> A e. RR* )
11	4	simp1d 1040	. . . . . 6 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> B e. RR* )
12		ixxss1.3	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ w e. RR* ) -> ( ( A W B /\ B T w ) -> A R w ) )
13	10, 11, 5, 12	syl3anc 1278	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> ( ( A W B /\ B T w ) -> A R w ) )
14	6, 9, 13	mp2and 437	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> A R w )
15	8	simprd 114	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> w S C )
16	4	simp2d 1041	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> C 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 10299	. . . . 5 |- ( ( A e. RR* /\ C e. RR* ) -> ( w e. ( A O C ) <-> ( w e. RR* /\ A R w /\ w S C ) ) )
19	10, 16, 18	syl2anc 415	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> ( w e. ( A O C ) <-> ( w e. RR* /\ A R w /\ w S C ) ) )
20	5, 14, 15, 19	mpbir3and 1211	. . 3 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> w e. ( A O C ) )
21	20	ex 115	. 2 |- ( ( A e. RR* /\ A W B ) -> ( w e. ( B P C ) -> w e. ( A O C ) ) )
22	21	ssrdv 3254	1 |- ( ( A e. RR* /\ A W B ) -> ( B P C ) C_ ( A O C ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220   class class class wbr 4130  (class class class)co 6085   e. cmpo 6087   RR*cxr 8359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364

This theorem is used by:  iooss1  10318