Theorem dfnul2 3360

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 3359	. . . 4 ⊢ ∅ = (V ∖ V)
2	1	eleq2i 2204	. . 3 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3		eldif 3075	. . 3 ⊢ (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4		pm3.24 682	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
5		eqid 2137	. . . . 5 ⊢ 𝑥 = 𝑥
6	5	notnoti 634	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
7	4, 6	2false 690	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
8	2, 3, 7	3bitri 205	. 2 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
9	8	abbi2i 2252	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1331   ∈ wcel 1480  {cab 2123  Vcvv 2681   ∖ cdif 3063  ∅c0 3358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-nul 3359

This theorem is referenced by:  dfnul3  3361  rab0  3386  iotanul  5098