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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
StepHypRef Expression
1 equid 1926 . . . . 5 𝑥 = 𝑥
2 noel 3878 . . . . . 6 ¬ 𝑥 ∈ ∅
32intnanr 952 . . . . 5 ¬ (𝑥 ∈ ∅ ∧ 𝜑)
41, 32th 253 . . . 4 (𝑥 = 𝑥 ↔ ¬ (𝑥 ∈ ∅ ∧ 𝜑))
54con2bii 346 . . 3 ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
65abbii 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}
96, 7, 83eqtr4i 2642 1 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
 Colors of variables: wff setvar class 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)
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