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Theorem elixx3g 8540
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  RR* and  RR*. (Contributed by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
ixxssxr.1  O  RR* ,  RR*  |->  {  RR*  |  R  S }
Assertion
Ref Expression
elixx3g  C  O  RR*  RR*  C 
RR*  R C  C S
Distinct variable groups:   ,,, R   , S,,   ,,,   ,,,   , C,,
Allowed substitution hints:    O(,,)

Proof of Theorem elixx3g
StepHypRef Expression
1 anass 381 . 2  RR*  RR*  C  RR*  R C  C S  RR*  RR*  C  RR*  R C  C S
2 df-3an 886 . . 3  RR*  RR*  C 
RR*  RR*  RR*  C  RR*
32anbi1i 431 . 2  RR*  RR*  C  RR*  R C  C S  RR*  RR*  C  RR*  R C  C S
4 ixxssxr.1 . . . 4  O  RR* ,  RR*  |->  {  RR*  |  R  S }
54elmpt2cl 5640 . . 3  C  O  RR*  RR*
64elixx1 8536 . . . 4  RR*  RR*  C  O  C  RR*  R C  C S
7 3anass 888 . . . 4  C  RR*  R C  C S  C  RR*  R C  C S
86, 7syl6bb 185 . . 3  RR*  RR*  C  O  C  RR*  R C  C S
95, 8biadan2 429 . 2  C  O  RR*  RR*  C  RR*  R C  C S
101, 3, 93bitr4ri 202 1  C  O  RR*  RR*  C 
RR*  R C  C S
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390   {crab 2304   class class class wbr 3755  (class class class)co 5455    |-> cmpt2 5457   RR*cxr 6856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-pnf 6859  df-mnf 6860  df-xr 6861
This theorem is referenced by:  ixxss1  8543  ixxss2  8544  ixxss12  8545  elioo3g  8549  iccss2  8583  iccssico2  8586
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