MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfnul2 Structured version   Visualization version   GIF version

Theorem dfnul2 4179
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 2077, ax-11 2091, and ax-12 2104. (Revised by Steven Nguyen, 3-May-2023.)
Assertion
Ref Expression
dfnul2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem dfnul2
StepHypRef Expression
1 df-nul 4178 . 2 ∅ = (V ∖ V)
2 df-dif 3831 . 2 (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3 pm3.24 394 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4 equid 1968 . . . . 5 𝑥 = 𝑥
54notnoti 140 . . . 4 ¬ ¬ 𝑥 = 𝑥
63, 52false 368 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
76abbii 2841 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
81, 2, 73eqtri 2803 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 387   = wceq 1507  wcel 2048  {cab 2755  Vcvv 3412  cdif 3825  c0 4177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-9 2057  ax-ext 2747
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-sb 2014  df-clab 2756  df-cleq 2768  df-dif 3831  df-nul 4178
This theorem is referenced by:  dfnul3  4181  iotanul  6165  avril1  28013
  Copyright terms: Public domain W3C validator