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Theorem elioo4g 10800
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 10798 . . . . 5  |-  ( C  e.  ( A (,) B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
2 elioore 10775 . . . . 5  |-  ( C  e.  ( A (,) B )  ->  C  e.  RR )
31, 2jca 518 . . . 4  |-  ( C  e.  ( A (,) B )  ->  (
( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR ) )
4 df-3an 936 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR ) )
53, 4sylibr 203 . . 3  |-  ( C  e.  ( A (,) B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR ) )
6 eliooord 10799 . . 3  |-  ( C  e.  ( A (,) B )  ->  ( A  <  C  /\  C  <  B ) )
75, 6jca 518 . 2  |-  ( C  e.  ( A (,) B )  ->  (
( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) ) )
8 rexr 8964 . . . . 5  |-  ( C  e.  RR  ->  C  e.  RR* )
983anim3i 1139 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* ) )
109anim1i 551 . . 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 10774 . . 3  |-  ( C  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  C  <  B ) ) )
1210, 11sylibr 203 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) )  ->  C  e.  ( A (,) B ) )
137, 12impbii 180 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:    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1710   class class class wbr 4102  (class class class)co 5942   RRcr 8823   RR*cxr 8953    < clt 8954   (,)cioo 10745
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 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-pre-lttri 8898  ax-pre-lttrn 8899
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2524  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-ioo 10749
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