Theorem bj-dfnul2 37271

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 2178, ax-11 2194, and ax-12 2213. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dfnul2	⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem bj-dfnul2

Step	Hyp	Ref	Expression
1		dfnul4 4281	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
2		equid 2045	. . . 4 ⊢ 𝑥 = 𝑥
3	2	bj-ntrufal 37270	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥)
4	3	abbii 2827	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
5	1, 4	eqtr4i 2786	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ⊥wfal 1582  {cab 2738  ∅c0 4279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-dif 3902  df-nul 4280

This theorem is used by: (None)