Theorem dfnul2 3392

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 3391	. . . 4 ⊢ ∅ = (V ∖ V)
2	1	eleq2i 2221	. . 3 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3		eldif 3107	. . 3 ⊢ (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4		pm3.24 683	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
5		eqid 2154	. . . . 5 ⊢ 𝑥 = 𝑥
6	5	notnoti 635	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
7	4, 6	2false 691	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
8	2, 3, 7	3bitri 205	. 2 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
9	8	abbi2i 2269	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1332   ∈ wcel 2125  {cab 2140  Vcvv 2709   ∖ cdif 3095  ∅c0 3390

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100  df-nul 3391

This theorem is referenced by:  dfnul3  3393  rab0  3418  iotanul  5143