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

Step	Hyp	Ref	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
5	4	notnoti 140	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
6	3, 5	2false 368	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
7	6	abbii 2841	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
8	1, 2, 7	3eqtri 2803	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

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