Theorem dfnul3 3918

Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)

Assertion

Ref	Expression
dfnul3	⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}

Proof of Theorem dfnul3

Step	Hyp	Ref	Expression
1		pm3.24 926	. . . . 5 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)
2		equid 1939	. . . . 5 ⊢ 𝑥 = 𝑥
3	1, 2	2th 254	. . . 4 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴) ↔ 𝑥 = 𝑥)
4	3	con1bii 346	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴))
5	4	abbii 2739	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)}
6		dfnul2 3917	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7		df-rab 2921	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)}
8	5, 6, 7	3eqtr4i 2654	1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}

Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  {crab 2916  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  difidALT  3949  kmlem3  8974