Theorem dfnul2 3220

Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)

Assertion

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

Proof of Theorem dfnul2

Step	Hyp	Ref	Expression
1		df-nul 3219	. . . 4 ⊢ ∅ = (V ∖ V)
2	1	eleq2i 2101	. . 3 ⊢ (x ∈ ∅ ↔ x ∈ (V ∖ V))
3		eldif 2921	. . 3 ⊢ (x ∈ (V ∖ V) ↔ (x ∈ V ∧ ¬ x ∈ V))
4		pm3.24 626	. . . 4 ⊢ ¬ (x ∈ V ∧ ¬ x ∈ V)
5		eqid 2037	. . . . 5 ⊢ x = x
6	5	notnoti 573	. . . 4 ⊢ ¬ ¬ x = x
7	4, 6	2false 616	. . 3 ⊢ ((x ∈ V ∧ ¬ x ∈ V) ↔ ¬ x = x)
8	2, 3, 7	3bitri 195	. 2 ⊢ (x ∈ ∅ ↔ ¬ x = x)
9	8	abbi2i 2149	1 ⊢ ∅ = {x ∣ ¬ x = x}

Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1242   ∈ wcel 1390  {cab 2023  Vcvv 2551   ∖ cdif 2908  ∅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-v 2553  df-dif 2914  df-nul 3219

This theorem is referenced by:  dfnul3  3221  rab0  3240  iotanul  4825