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Theorem elixx3g 9845
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
ixxssxr.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, R    x, S, y, z    x, A, y, z    x, B, y, z    x, C, y, z
Allowed substitution hints:    O( x, y, z)

Proof of Theorem elixx3g
StepHypRef Expression
1 anass 399 . 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 975 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR* ) )
32anbi1i 455 . 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 ixxssxr.1 . . . 4  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
54elmpocl 6044 . . 3  |-  ( C  e.  ( A O B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
64elixx1 9841 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  ( C  e.  RR*  /\  A R C  /\  C S B ) ) )
7 3anass 977 . . . 4  |-  ( ( C  e.  RR*  /\  A R C  /\  C S B )  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
86, 7bitrdi 195 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) )
95, 8biadan2 453 . 2  |-  ( C  e.  ( A O B )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) )
101, 3, 93bitr4ri 212 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:    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   {crab 2452   class class class wbr 3987  (class class class)co 5850    e. cmpo 5852   RR*cxr 7940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-xr 7945
This theorem is referenced by:  ixxss1  9848  ixxss2  9849  ixxss12  9850  elioo3g  9854  iccss2  9888  iccssico2  9891  elicore  10210
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