Theorem rab0OLD 4335

Description: Obsolete version of rab0 4334 as of 10-Jun-2026. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rab0OLD	⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Proof of Theorem rab0OLD

Step	Hyp	Ref	Expression
1		df-rab 3413	. 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
2		ab0 4328	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑))
3		noel 4283	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
4	3	intnanr 493	. . 3 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑)
5	2, 4	mpgbir 1831	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅
6	1, 5	eqtri 2783	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  {crab 3412  ∅c0 4278

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-8 2146  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-clel 2835  df-rab 3413  df-dif 4047  df-nul 4279

This theorem is used by: (None)