Theorem dfnul3 3221

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		equid 1586	. . . . 5 ⊢ x = x
2	1	notnoti 573	. . . 4 ⊢ ¬ ¬ x = x
3		pm3.24 626	. . . 4 ⊢ ¬ (x ∈ A ∧ ¬ x ∈ A)
4	2, 3	2false 616	. . 3 ⊢ (¬ x = x ↔ (x ∈ A ∧ ¬ x ∈ A))
5	4	abbii 2150	. 2 ⊢ {x ∣ ¬ x = x} = {x ∣ (x ∈ A ∧ ¬ x ∈ A)}
6		dfnul2 3220	. 2 ⊢ ∅ = {x ∣ ¬ x = x}
7		df-rab 2309	. 2 ⊢ {x ∈ A ∣ ¬ x ∈ A} = {x ∣ (x ∈ A ∧ ¬ x ∈ A)}
8	5, 6, 7	3eqtr4i 2067	1 ⊢ ∅ = {x ∈ A ∣ ¬ x ∈ A}

Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1242   ∈ wcel 1390  {cab 2023  {crab 2304  ∅c0 3218

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-nul 3219

This theorem is referenced by:  difidALT  3287