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Theorem rab0OLD 3936
Description: Obsolete proof of rab0 3935 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 1936 . . . . 5 𝑥 = 𝑥
2 noel 3901 . . . . . 6 ¬ 𝑥 ∈ ∅
32intnanr 960 . . . . 5 ¬ (𝑥 ∈ ∅ ∧ 𝜑)
41, 32th 254 . . . 4 (𝑥 = 𝑥 ↔ ¬ (𝑥 ∈ ∅ ∧ 𝜑))
54con2bii 347 . . 3 ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
65abbii 2736 . 2 {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 2917 . 2 {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
8 dfnul2 3899 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
96, 7, 83eqtr4i 2653 1 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  {cab 2607  {crab 2912  c0 3897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-nul 3898
This theorem is referenced by: (None)
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