MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elixx1 Unicode version

Theorem elixx1 10757
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 10756 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  =  { z  e.  RR*  |  ( A R z  /\  z S B ) } )
32eleq2d 2425 . 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 4108 . . . . 5  |-  ( z  =  C  ->  ( A R z  <->  A R C ) )
5 breq1 4107 . . . . 5  |-  ( z  =  C  ->  (
z S B  <->  C S B ) )
64, 5anbi12d 691 . . . 4  |-  ( z  =  C  ->  (
( A R z  /\  z S B )  <->  ( A R C  /\  C S B ) ) )
76elrab 2999 . . 3  |-  ( C  e.  { z  e. 
RR*  |  ( A R z  /\  z S B ) }  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
8 3anass 938 . . 3  |-  ( ( C  e.  RR*  /\  A R C  /\  C S B )  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
97, 8bitr4i 243 . 2  |-  ( C  e.  { z  e. 
RR*  |  ( A R z  /\  z S B ) }  <->  ( C  e.  RR*  /\  A R C  /\  C S B ) )
103, 9syl6bb 252 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   {crab 2623   class class class wbr 4104  (class class class)co 5945    e. cmpt2 5947   RR*cxr 8956
This theorem is referenced by:  elixx3g  10761  ixxssixx  10762  ixxdisj  10763  ixxun  10764  ixxss1  10766  ixxss2  10767  ixxss12  10768  ixxub  10769  ixxlb  10770  elioo1  10788  elioc1  10790  elico1  10791  elicc1  10792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-xr 8961
  Copyright terms: Public domain W3C validator