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Theorem ab0orv 4152
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 2010 . 2 𝑥𝜑
2 dfnf5 4151 . 2 (Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
31, 2mpbi 222 1 ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅)
Colors of variables: wff setvar class
Syntax hints:  wo 874   = wceq 1653  wnf 1879  {cab 2783  Vcvv 3383  c0 4113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-v 3385  df-dif 3770  df-nul 4114
This theorem is referenced by: (None)
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