Theorem dfnul2 3270

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 3269	. . . 4 ⊢ ∅ = (V ∖ V)
2	1	eleq2i 2149	. . 3 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3		eldif 2992	. . 3 ⊢ (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4		pm3.24 660	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
5		eqid 2083	. . . . 5 ⊢ 𝑥 = 𝑥
6	5	notnoti 607	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
7	4, 6	2false 650	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
8	2, 3, 7	3bitri 204	. 2 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
9	8	abbi2i 2197	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   ∧ wa 102   = wceq 1285   ∈ wcel 1434  {cab 2069  Vcvv 2611   ∖ cdif 2980  ∅c0 3268

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613  df-dif 2985  df-nul 3269

This theorem is referenced by:  dfnul3  3271  rab0  3290  iotanul  4933