Theorem dfnul2 3452

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 3451	. . . 4 ⊢ ∅ = (V ∖ V)
2	1	eleq2i 2263	. . 3 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3		eldif 3166	. . 3 ⊢ (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4		pm3.24 694	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
5		eqid 2196	. . . . 5 ⊢ 𝑥 = 𝑥
6	5	notnoti 646	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
7	4, 6	2false 702	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
8	2, 3, 7	3bitri 206	. 2 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
9	8	abbi2i 2311	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1364   ∈ wcel 2167  {cab 2182  Vcvv 2763   ∖ cdif 3154  ∅c0 3450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-nul 3451

This theorem is referenced by:  dfnul3  3453  rab0  3479  iotanul  5234