Theorem ab0orv 3953

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 1843	. 2 ⊢ Ⅎ𝑥𝜑
2		dfnf5 3952	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V))
3	1, 2	mpbi 220	1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V)

Syntax hints:   ∨ wo 383   = wceq 1483  Ⅎwnf 1708  {cab 2608  Vcvv 3200  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)