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Theorem elioo4g 9657
Description: Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
elioo4g  |-  ( C  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) ) )

Proof of Theorem elioo4g
StepHypRef Expression
1 eliooxr 9650 . . . . 5  |-  ( C  e.  ( A (,) B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
2 elioore 9635 . . . . 5  |-  ( C  e.  ( A (,) B )  ->  C  e.  RR )
31, 2jca 302 . . . 4  |-  ( C  e.  ( A (,) B )  ->  (
( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR ) )
4 df-3an 947 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR ) )
53, 4sylibr 133 . . 3  |-  ( C  e.  ( A (,) B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR ) )
6 eliooord 9651 . . 3  |-  ( C  e.  ( A (,) B )  ->  ( A  <  C  /\  C  <  B ) )
75, 6jca 302 . 2  |-  ( C  e.  ( A (,) B )  ->  (
( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) ) )
8 rexr 7775 . . . . 5  |-  ( C  e.  RR  ->  C  e.  RR* )
983anim3i 1152 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* ) )
109anim1i 336 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) )  -> 
( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  C  <  B ) ) )
11 elioo3g 9633 . . 3  |-  ( C  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  C  <  B ) ) )
1210, 11sylibr 133 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) )  ->  C  e.  ( A (,) B ) )
137, 12impbii 125 1  |-  ( C  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 945    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   RRcr 7583   RR*cxr 7763    < clt 7764   (,)cioo 9611
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-po 4186  df-iso 4187  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-ioo 9615
This theorem is referenced by: (None)
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