Theorem rab0OLD 3910

Description: Obsolete proof of rab0 3909 as of 14-Jul-2021. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem rab0OLD

Step	Hyp	Ref	Expression
1		equid 1926	. . . . 5 ⊢ 𝑥 = 𝑥
2		noel 3878	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
3	2	intnanr 952	. . . . 5 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑)
4	1, 3	2th 253	. . . 4 ⊢ (𝑥 = 𝑥 ↔ ¬ (𝑥 ∈ ∅ ∧ 𝜑))
5	4	con2bii 346	. . 3 ⊢ ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
6	5	abbii 2726	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7		df-rab 2905	. 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
8		dfnul2 3876	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
9	6, 7, 8	3eqtr4i 2642	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  ∅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-rab 2905  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by: (None)