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Theorem brabv 6323
 Description: If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
Assertion
Ref Expression
brabv

Proof of Theorem brabv
StepHypRef Expression
1 df-br 4396 . 2
2 opprc 4181 . . . 4
3 0neqopab 6322 . . . . 5
4 eleq1 2474 . . . . 5
53, 4mtbiri 301 . . . 4
62, 5syl 17 . . 3
76con4i 130 . 2
81, 7sylbi 195 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 367   wceq 1405   wcel 1842  cvv 3059  c0 3738  cop 3978   class class class wbr 4395  copab 4452 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454 This theorem is referenced by:  bropopvvv  6864  isfunc  15477  eqgval  16574
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