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Theorem elixx3g 10929
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 10925 . . . 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 10926 . . . . . . 7  |-  O :
( RR*  X.  RR* ) --> ~P RR*
1110fdmi 5596 . . . . . 6  |-  dom  O  =  ( RR*  X.  RR* )
1211ndmov 6231 . . . . 5  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  =  (/) )
1312eleq2d 2503 . . . 4  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <-> 
C  e.  (/) ) )
14 noel 3632 . . . . . 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 1652    e. wcel 1725   {crab 2709   (/)c0 3628   ~Pcpw 3799   class class class wbr 4212    X. cxp 4876  (class class class)co 6081    e. cmpt2 6083   RR*cxr 9119
This theorem is referenced by:  ixxss1  10934  ixxss2  10935  ixxss12  10936  elioo3g  10945  iccss2  10981  iccssico2  10984  xrtgioo  18837  ftc1anclem7  26286  ftc1anclem8  26287  ftc1anc  26288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-xr 9124
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