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Theorem reapval 7340
 Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7352 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 3760 . . . 4
2 simpr 103 . . . . 5
3 simpl 102 . . . . 5
42, 3breq12d 3768 . . . 4
51, 4orbi12d 706 . . 3
6 df-reap 7339 . . 3 #
75, 6brab2ga 4358 . 2 #
87baib 827 1 #
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wo 628   wceq 1242   wcel 1390   class class class wbr 3755  cr 6690   clt 6837   #ℝ creap 7338 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-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-bnd 1396  ax-4 1397  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 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  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-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-reap 7339 This theorem is referenced by:  reapirr  7341  recexre  7342  reapti  7343  reaplt  7352
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