Theorem ab0orvALT 4332

Description: Alternate proof of ab0orv 4331, shorter but using more axioms. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ab0orvALT	⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)

Distinct variable group:   𝜑,𝑥

Proof of Theorem ab0orvALT

Step	Hyp	Ref	Expression
1		nfv 1946	. 2 ⊢ Ⅎ𝑥𝜑
2		dfnf5 4330	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))
3	1, 2	mpbi 233	1 ⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1569  Ⅎwnf 1815  {cab 2738  Vcvv 3450  ∅c0 4277

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-v 3452  df-dif 4046  df-nul 4278

This theorem is used by: (None)