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Theorem dfnul3 4292
Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
dfnul3 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}

Proof of Theorem dfnul3
StepHypRef Expression
1 pm3.24 403 . . . . 5 ¬ (𝑥𝐴 ∧ ¬ 𝑥𝐴)
2 equid 2010 . . . . 5 𝑥 = 𝑥
31, 22th 265 . . . 4 (¬ (𝑥𝐴 ∧ ¬ 𝑥𝐴) ↔ 𝑥 = 𝑥)
43con1bii 358 . . 3 𝑥 = 𝑥 ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐴))
54abbii 2883 . 2 {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
6 dfnul2 4290 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 3144 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐴} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
85, 6, 73eqtr4i 2851 1 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1528  wcel 2105  {cab 2796  {crab 3139  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-rab 3144  df-dif 3936  df-nul 4289
This theorem is referenced by:  difidALT  4328  kmlem3  9566
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