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	⊢ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V)

Distinct variable group:   𝜑,𝑥

Proof of Theorem ab0orv

Step	Hyp	Ref	Expression
1		nfv 1883	. 2 ⊢ Ⅎ𝑥𝜑
2		dfnf5 3985	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V))
3	1, 2	mpbi 220	1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V)

Syntax hints:   ∨ wo 382   = wceq 1523  Ⅎwnf 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)