MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elixx1 Unicode version

Theorem elixx1 10889
Description: Membership in an interval of extended reals. (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
elixx1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  ( 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 elixx1
StepHypRef Expression
1 ixx.1 . . . 4  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
21ixxval 10888 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  =  { z  e.  RR*  |  ( A R z  /\  z S B ) } )
32eleq2d 2479 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  C  e.  { z  e.  RR*  |  ( A R z  /\  z S B ) } ) )
4 breq2 4184 . . . . 5  |-  ( z  =  C  ->  ( A R z  <->  A R C ) )
5 breq1 4183 . . . . 5  |-  ( z  =  C  ->  (
z S B  <->  C S B ) )
64, 5anbi12d 692 . . . 4  |-  ( z  =  C  ->  (
( A R z  /\  z S B )  <->  ( A R C  /\  C S B ) ) )
76elrab 3060 . . 3  |-  ( C  e.  { z  e. 
RR*  |  ( A R z  /\  z S B ) }  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
8 3anass 940 . . 3  |-  ( ( C  e.  RR*  /\  A R C  /\  C S B )  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
97, 8bitr4i 244 . 2  |-  ( C  e.  { z  e. 
RR*  |  ( A R z  /\  z S B ) }  <->  ( C  e.  RR*  /\  A R C  /\  C S B ) )
103, 9syl6bb 253 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  ( C  e.  RR*  /\  A R C  /\  C S B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {crab 2678   class class class wbr 4180  (class class class)co 6048    e. cmpt2 6050   RR*cxr 9083
This theorem is referenced by:  elixx3g  10893  ixxssixx  10894  ixxdisj  10895  ixxun  10896  ixxss1  10898  ixxss2  10899  ixxss12  10900  ixxub  10901  ixxlb  10902  elioo1  10920  elioc1  10922  elico1  10923  elicc1  10924
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-xr 9088
  Copyright terms: Public domain W3C validator