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Theorem rab0OLD 4099
Description: Obsolete proof of rab0 4098 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 2094 . . . . 5 𝑥 = 𝑥
2 noel 4062 . . . . . 6 ¬ 𝑥 ∈ ∅
32intnanr 999 . . . . 5 ¬ (𝑥 ∈ ∅ ∧ 𝜑)
41, 32th 254 . . . 4 (𝑥 = 𝑥 ↔ ¬ (𝑥 ∈ ∅ ∧ 𝜑))
54con2bii 346 . . 3 ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
65abbii 2877 . 2 {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 3059 . 2 {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
8 dfnul2 4060 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
96, 7, 83eqtr4i 2792 1 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1632  wcel 2139  {cab 2746  {crab 3054  c0 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-nul 4059
This theorem is referenced by: (None)
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