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Theorem dfnul2 4240
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 2141, ax-11 2158, 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 4239 . 2 ∅ = {𝑥 ∣ ⊥}
2 equid 2020 . . . . 5 𝑥 = 𝑥
32notnoti 145 . . . 4 ¬ ¬ 𝑥 = 𝑥
43bifal 1559 . . 3 𝑥 = 𝑥 ↔ ⊥)
54abbii 2808 . 2 {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
61, 5eqtr4i 2768 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1543  wfal 1555  {cab 2714  c0 4237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-dif 3869  df-nul 4238
This theorem is referenced by:  dfnul3OLD  4243  dfnul4OLD  4244  ab0orv  4293  iotanul  6358  avril1  28546
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