Theorem dfnul2 3553

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 3552	. . . 4 ⊢ ∅ = (V ∖ V)
2	1	eleq2i 2417	. . 3 ⊢ (x ∈ ∅ ↔ x ∈ (V ∖ V))
3		eldif 3222	. . 3 ⊢ (x ∈ (V ∖ V) ↔ (x ∈ V ∧ ¬ x ∈ V))
4		eqid 2353	. . . . 5 ⊢ x = x
5		pm3.24 852	. . . . 5 ⊢ ¬ (x ∈ V ∧ ¬ x ∈ V)
6	4, 5	2th 230	. . . 4 ⊢ (x = x ↔ ¬ (x ∈ V ∧ ¬ x ∈ V))
7	6	con2bii 322	. . 3 ⊢ ((x ∈ V ∧ ¬ x ∈ V) ↔ ¬ x = x)
8	2, 3, 7	3bitri 262	. 2 ⊢ (x ∈ ∅ ↔ ¬ x = x)
9	8	eqabi 2465	1 ⊢ ∅ = {x ∣ ¬ x = x}

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860   ∖ cdif 3207  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  dfnul3  3554  rab0  3572  iotanul  4355