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Theorem elioo4g 9321
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 9314 . . . . 5  |-  ( C  e.  ( A (,) B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
2 elioore 9299 . . . . 5  |-  ( C  e.  ( A (,) B )  ->  C  e.  RR )
31, 2jca 300 . . . 4  |-  ( C  e.  ( A (,) B )  ->  (
( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR ) )
4 df-3an 926 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR ) )
53, 4sylibr 132 . . 3  |-  ( C  e.  ( A (,) B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR ) )
6 eliooord 9315 . . 3  |-  ( C  e.  ( A (,) B )  ->  ( A  <  C  /\  C  <  B ) )
75, 6jca 300 . 2  |-  ( C  e.  ( A (,) B )  ->  (
( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) ) )
8 rexr 7512 . . . . 5  |-  ( C  e.  RR  ->  C  e.  RR* )
983anim3i 1131 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* ) )
109anim1i 333 . . 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 9297 . . 3  |-  ( C  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  C  <  B ) ) )
1210, 11sylibr 132 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) )  ->  C  e.  ( A (,) B ) )
137, 12impbii 124 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 102    <-> wb 103    /\ w3a 924    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   RRcr 7328   RR*cxr 7500    < clt 7501   (,)cioo 9275
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-ioo 9279
This theorem is referenced by: (None)
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