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Theorem ab0orv 4332
Description: The class builder of a wff not containing the abstraction variable is either the empty set or the universal class. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.)
Assertion
Ref Expression
ab0orv ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅)
Distinct variable group:   𝜑,𝑥

Proof of Theorem ab0orv
StepHypRef Expression
1 nfv 1906 . 2 𝑥𝜑
2 dfnf5 4331 . 2 (Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
31, 2mpbi 231 1 ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅)
Colors of variables: wff setvar class
Syntax hints:  wo 841   = wceq 1528  wnf 1775  {cab 2796  Vcvv 3492  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936  df-nul 4289
This theorem is referenced by: (None)
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