Theorem dfnul2 1709

Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20.

Assertion

Ref	Expression
dfnul2	⊢ ∅ = {x∣ ¬ x = x}

Proof of Theorem dfnul2

Step	Hyp	Ref	Expression
1		df-nul 1708	. . . 4 ⊢ ∅ = (V ∖ V)
2	1	eleq2i 1153	. . 3 ⊢ (x ∈ ∅ ↔ x ∈ (V ∖ V))
3		eldif 1496	. . 3 ⊢ (x ∈ (V ∖ V) ↔ (x ∈ V ∧ ¬ x ∈ V))
4		cleqid 1102	. . . . 5 ⊢ x = x
5		pm3.24 496	. . . . 5 ⊢ ¬ (x ∈ V ∧ ¬ x ∈ V)
6	4, 5	2th 540	. . . 4 ⊢ (x = x ↔ ¬ (x ∈ V ∧ ¬ x ∈ V))
7	6	bicon2i 194	. . 3 ⊢ ((x ∈ V ∧ ¬ x ∈ V) ↔ ¬ x = x)
8	2, 3, 7	3bitr 155	. 2 ⊢ (x ∈ ∅ ↔ ¬ x = x)
9	8	biabri 1180	1 ⊢ ∅ = {x∣ ¬ x = x}

Syntax hints:  ¬ wn 1   ∧ wa 196   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484  ∅c0 1707

This theorem is referenced by:  dfnul3 1710  noel 1711  dm0 2542  dmsn0 2543  dmsnsn0 2544

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-nul 1708

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