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Theorem ab0orv 3986
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} = ∅ ∨ {𝑥𝜑} = V)
Distinct variable group:   𝜑,𝑥

Proof of Theorem ab0orv
StepHypRef Expression
1 nfv 1883 . 2 𝑥𝜑
2 dfnf5 3985 . 2 (Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = ∅ ∨ {𝑥𝜑} = V))
31, 2mpbi 220 1 ({𝑥𝜑} = ∅ ∨ {𝑥𝜑} = V)
Colors of variables: wff setvar class
Syntax hints:  wo 382   = wceq 1523  wnf 1748  {cab 2637  Vcvv 3231  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-nul 3949
This theorem is referenced by: (None)
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