ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elixx1 Unicode version

Theorem elixx1 9066
Description: Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
Assertion
Ref Expression
elixx1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  ( C  e.  RR*  /\  A R C  /\  C S B ) ) )
Distinct variable groups:    x, y, z, A    x, C, y, z    x, B, y, z    x, R, y, z    x, S, y, z
Allowed substitution hints:    O( x, y, z)

Proof of Theorem elixx1
StepHypRef Expression
1 ixx.1 . . . 4  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
21ixxval 9065 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  =  { z  e.  RR*  |  ( A R z  /\  z S B ) } )
32eleq2d 2152 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  C  e.  { z  e.  RR*  |  ( A R z  /\  z S B ) } ) )
4 breq2 3809 . . . . 5  |-  ( z  =  C  ->  ( A R z  <->  A R C ) )
5 breq1 3808 . . . . 5  |-  ( z  =  C  ->  (
z S B  <->  C S B ) )
64, 5anbi12d 457 . . . 4  |-  ( z  =  C  ->  (
( A R z  /\  z S B )  <->  ( A R C  /\  C S B ) ) )
76elrab 2757 . . 3  |-  ( C  e.  { z  e. 
RR*  |  ( A R z  /\  z S B ) }  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
8 3anass 924 . . 3  |-  ( ( C  e.  RR*  /\  A R C  /\  C S B )  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
97, 8bitr4i 185 . 2  |-  ( C  e.  { z  e. 
RR*  |  ( A R z  /\  z S B ) }  <->  ( C  e.  RR*  /\  A R C  /\  C S B ) )
103, 9syl6bb 194 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  ( C  e.  RR*  /\  A R C  /\  C S B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   {crab 2357   class class class wbr 3805  (class class class)co 5564    |-> cmpt2 5566   RR*cxr 7284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7199  ax-resscn 7200
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7287  df-mnf 7288  df-xr 7289
This theorem is referenced by:  elixx3g  9070  ixxssixx  9071  ixxdisj  9072  ixxss1  9073  ixxss2  9074  ixxss12  9075  elioo1  9080  elioc1  9091  elico1  9092  elicc1  9093
  Copyright terms: Public domain W3C validator