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Theorem dfnul2 4342
Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2139, ax-11 2155, and ax-12 2175. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
Assertion
Ref Expression
dfnul2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem dfnul2
StepHypRef Expression
1 dfnul4 4341 . 2 ∅ = {𝑥 ∣ ⊥}
2 equid 2009 . . . . 5 𝑥 = 𝑥
32notnoti 143 . . . 4 ¬ ¬ 𝑥 = 𝑥
43bifal 1553 . . 3 𝑥 = 𝑥 ↔ ⊥)
54abbii 2807 . 2 {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
61, 5eqtr4i 2766 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wfal 1549  {cab 2712  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-dif 3966  df-nul 4340
This theorem is referenced by:  ab0orv  4389  iotanul  6541  avril1  30492
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