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Theorem elixx3g 9288
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 393 . 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 926 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR* ) )
32anbi1i 446 . 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 ) } )
54elmpt2cl 5824 . . 3  |-  ( C  e.  ( A O B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
64elixx1 9284 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  ( C  e.  RR*  /\  A R C  /\  C S B ) ) )
7 3anass 928 . . . 4  |-  ( ( C  e.  RR*  /\  A R C  /\  C S B )  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) )
86, 7syl6bb 194 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B )  <->  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) )
95, 8biadan2 444 . 2  |-  ( C  e.  ( A O B )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  ( A R C  /\  C S B ) ) ) )
101, 3, 93bitr4ri 211 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   {crab 2363   class class class wbr 3837  (class class class)co 5634    |-> cmpt2 5636   RR*cxr 7500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505
This theorem is referenced by:  ixxss1  9291  ixxss2  9292  ixxss12  9293  elioo3g  9297  iccss2  9331  iccssico2  9334
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