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Theorem reapval 7795
 Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7807 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
reapval #

Proof of Theorem reapval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 3810 . . . 4
2 simpr 108 . . . . 5
3 simpl 107 . . . . 5
42, 3breq12d 3818 . . . 4
51, 4orbi12d 740 . . 3
6 df-reap 7794 . . 3 #
75, 6brab2ga 4461 . 2 #
87baib 862 1 #
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103   wo 662   wceq 1285   wcel 1434   class class class wbr 3805  cr 7094   clt 7267   #ℝ creap 7793 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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-reap 7794 This theorem is referenced by:  reapirr  7796  recexre  7797  reapti  7798  reaplt  7807
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