Theorem dfnul2OLD 4296

Description: Obsolete version of dfnul2 4295 as of 3-May-2023. Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
dfnul2OLD	⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem dfnul2OLD

Step	Hyp	Ref	Expression
1		df-nul 4294	. . . 4 ⊢ ∅ = (V ∖ V)
2	1	eleq2i 2906	. . 3 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3		eldif 3948	. . 3 ⊢ (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4		eqid 2823	. . . . 5 ⊢ 𝑥 = 𝑥
5		pm3.24 405	. . . . 5 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
6	4, 5	2th 266	. . . 4 ⊢ (𝑥 = 𝑥 ↔ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
7	6	con2bii 360	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
8	2, 3, 7	3bitri 299	. 2 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
9	8	abbi2i 2955	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2801  Vcvv 3496   ∖ cdif 3935  ∅c0 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-dif 3941  df-nul 4294

This theorem is referenced by: (None)