Theorem dfnul3 3554

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

Assertion

Ref	Expression
dfnul3	|- (/) = {x e. A | -. x e. A}

Proof of Theorem dfnul3

Step	Hyp	Ref	Expression
1		pm3.24 852	. . . . 5 |- -. (x e. A /\ -. x e. A)
2		eqid 2353	. . . . 5 |- x = x
3	1, 2	2th 230	. . . 4 |- (-. (x e. A /\ -. x e. A) <-> x = x)
4	3	con1bii 321	. . 3 |- (-. x = x <-> (x e. A /\ -. x e. A))
5	4	abbii 2466	. 2 |- {x | -. x = x} = {x | (x e. A /\ -. x e. A)}
6		dfnul2 3553	. 2 |- (/) = {x | -. x = x}
7		df-rab 2624	. 2 |- {x e. A | -. x e. A} = {x | (x e. A /\ -. x e. A)}
8	5, 6, 7	3eqtr4i 2383	1 |- (/) = {x e. A | -. x e. A}

Syntax hints:  -. wn 3   /\ wa 358   = wceq 1642   e. 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