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Theorem elixx1 9673
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 9672 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  =  { z  e.  RR*  |  ( A R z  /\  z S B ) } )
32eleq2d 2207 . 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 3928 . . . . 5  |-  ( z  =  C  ->  ( A R z  <->  A R C ) )
5 breq1 3927 . . . . 5  |-  ( z  =  C  ->  (
z S B  <->  C S B ) )
64, 5anbi12d 464 . . . 4  |-  ( z  =  C  ->  (
( A R z  /\  z S B )  <->  ( A R C  /\  C S B ) ) )
76elrab 2835 . . 3  |-  ( C  e.  { z  e. 
RR*  |  ( A R z  /\  z S B ) }  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
8 3anass 966 . . 3  |-  ( ( C  e.  RR*  /\  A R C  /\  C S B )  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
97, 8bitr4i 186 . 2  |-  ( C  e.  { z  e. 
RR*  |  ( A R z  /\  z S B ) }  <->  ( C  e.  RR*  /\  A R C  /\  C S B ) )
103, 9syl6bb 195 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 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   {crab 2418   class class class wbr 3924  (class class class)co 5767    e. cmpo 5769   RR*cxr 7792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797
This theorem is referenced by:  elixx3g  9677  ixxssixx  9678  ixxdisj  9679  ixxss1  9680  ixxss2  9681  ixxss12  9682  elioo1  9687  elioc1  9698  elico1  9699  elicc1  9700
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