Theorem ab0orv 3907

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 1830	. 2 ⊢ Ⅎ𝑥𝜑
2		dfnf5 3906	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V))
3	1, 2	mpbi 219	1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V)

Syntax hints:   ∨ wo 382   = wceq 1475  Ⅎwnf 1699  {cab 2596  Vcvv 3173  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by: (None)