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Theorem dfnul3 3894
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 pm3.24 925 . . . . 5 ¬ (𝑥𝐴 ∧ ¬ 𝑥𝐴)
2 equid 1936 . . . . 5 𝑥 = 𝑥
31, 22th 254 . . . 4 (¬ (𝑥𝐴 ∧ ¬ 𝑥𝐴) ↔ 𝑥 = 𝑥)
43con1bii 346 . . 3 𝑥 = 𝑥 ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐴))
54abbii 2736 . 2 {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
6 dfnul2 3893 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 2916 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐴} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
85, 6, 73eqtr4i 2653 1 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  {cab 2607  {crab 2911  c0 3891
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 2916  df-v 3188  df-dif 3558  df-nul 3892
This theorem is referenced by:  difidALT  3923  kmlem3  8918
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