Theorem dfnul3 3554

Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)

Assertion

Ref	Expression
dfnul3	⊢ ∅ = {x ∈ A ∣ ¬ x ∈ A}

Proof of Theorem dfnul3

Step	Hyp	Ref	Expression
1		pm3.24 852	. . . . 5 ⊢ ¬ (x ∈ A ∧ ¬ x ∈ A)
2		eqid 2353	. . . . 5 ⊢ x = x
3	1, 2	2th 230	. . . 4 ⊢ (¬ (x ∈ A ∧ ¬ x ∈ A) ↔ x = x)
4	3	con1bii 321	. . 3 ⊢ (¬ x = x ↔ (x ∈ A ∧ ¬ x ∈ A))
5	4	abbii 2466	. 2 ⊢ {x ∣ ¬ x = x} = {x ∣ (x ∈ A ∧ ¬ x ∈ A)}
6		dfnul2 3553	. 2 ⊢ ∅ = {x ∣ ¬ x = x}
7		df-rab 2624	. 2 ⊢ {x ∈ A ∣ ¬ x ∈ A} = {x ∣ (x ∈ A ∧ ¬ x ∈ A)}
8	5, 6, 7	3eqtr4i 2383	1 ⊢ ∅ = {x ∈ A ∣ ¬ x ∈ A}

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619  ∅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-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  difidALT  3620