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Theorem elixx3g 10861
Description: Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show  A  e.  RR* and  B  e.  RR*. (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
elixx3g  |-  ( C  e.  ( A O B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  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 elixx3g
StepHypRef Expression
1 anass 631 . 2  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  ( A R C  /\  C S B ) )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) )
2 df-3an 938 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR* ) )
32anbi1i 677 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A R C  /\  C S B ) )  <->  ( (
( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR* )  /\  ( A R C  /\  C S B ) ) )
4 ixx.1 . . . . 5  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
54elixx1 10857 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  ( C  e.  RR*  /\  A R C  /\  C S B ) ) )
6 3anass 940 . . . . 5  |-  ( ( C  e.  RR*  /\  A R C  /\  C S B )  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
7 ibar 491 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  ( A R C  /\  C S B ) )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) ) )
86, 7syl5bb 249 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A R C  /\  C S B )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) ) )
95, 8bitrd 245 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) ) )
104ixxf 10858 . . . . . . 7  |-  O :
( RR*  X.  RR* ) --> ~P RR*
1110fdmi 5536 . . . . . 6  |-  dom  O  =  ( RR*  X.  RR* )
1211ndmov 6170 . . . . 5  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  =  (/) )
1312eleq2d 2454 . . . 4  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <-> 
C  e.  (/) ) )
14 noel 3575 . . . . . 6  |-  -.  C  e.  (/)
1514pm2.21i 125 . . . . 5  |-  ( C  e.  (/)  ->  ( A  e.  RR*  /\  B  e. 
RR* ) )
16 simpl 444 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )  -> 
( A  e.  RR*  /\  B  e.  RR* )
)
1715, 16pm5.21ni 342 . . . 4  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  (/)  <->  (
( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) ) )
1813, 17bitrd 245 . . 3  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <-> 
( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) ) )
199, 18pm2.61i 158 . 2  |-  ( C  e.  ( A O B )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) )
201, 3, 193bitr4ri 270 1  |-  ( C  e.  ( A O B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A R C  /\  C S B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2653   (/)c0 3571   ~Pcpw 3742   class class class wbr 4153    X. cxp 4816  (class class class)co 6020    e. cmpt2 6022   RR*cxr 9052
This theorem is referenced by:  ixxss1  10866  ixxss2  10867  ixxss12  10868  elioo3g  10877  iccss2  10913  iccssico2  10916  xrtgioo  18708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-xr 9057
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