Theorem dfnul2 3552

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 3551	. . . 4 |- (/) = (_V \ _V)
2	1	eleq2i 2417	. . 3 |- (x e. (/) <-> x e. (_V \ _V))
3		eldif 3221	. . 3 |- (x e. (_V \ _V) <-> (x e. _V /\ -. x e. _V))
4		eqid 2353	. . . . 5 |- x = x
5		pm3.24 852	. . . . 5 |- -. (x e. _V /\ -. x e. _V)
6	4, 5	2th 230	. . . 4 |- (x = x <-> -. (x e. _V /\ -. x e. _V))
7	6	con2bii 322	. . 3 |- ((x e. _V /\ -. x e. _V) <-> -. x = x)
8	2, 3, 7	3bitri 262	. 2 |- (x e. (/) <-> -. x = x)
9	8	abbi2i 2464	1 |- (/) = {x | -. x = x}

Syntax hints:  -. wn 3   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  _Vcvv 2859   \ cdif 3206  (/)c0 3550

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by:  dfnul3  3553  rab0  3571  iotanul  4354