MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elioo4g Structured version   Unicode version

Theorem elioo4g 11585
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 11583 . . . . 5  |-  ( C  e.  ( A (,) B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
2 elioore 11559 . . . . 5  |-  ( C  e.  ( A (,) B )  ->  C  e.  RR )
31, 2jca 532 . . . 4  |-  ( C  e.  ( A (,) B )  ->  (
( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR ) )
4 df-3an 975 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR ) )
53, 4sylibr 212 . . 3  |-  ( C  e.  ( A (,) B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR ) )
6 eliooord 11584 . . 3  |-  ( C  e.  ( A (,) B )  ->  ( A  <  C  /\  C  <  B ) )
75, 6jca 532 . 2  |-  ( C  e.  ( A (,) B )  ->  (
( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) ) )
8 rexr 9639 . . . . 5  |-  ( C  e.  RR  ->  C  e.  RR* )
983anim3i 1184 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* ) )
109anim1i 568 . . 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 11558 . . 3  |-  ( C  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  C  <  B ) ) )
1210, 11sylibr 212 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) )  ->  C  e.  ( A (,) B ) )
137, 12impbii 188 1  |-  ( C  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767   class class class wbr 4447  (class class class)co 6284   RRcr 9491   RR*cxr 9627    < clt 9628   (,)cioo 11529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-ioo 11533
This theorem is referenced by:  cncfiooicclem1  31260  fourierdlem89  31524  fourierdlem90  31525  fourierdlem91  31526  fourierdlem100  31535
  Copyright terms: Public domain W3C validator