Theorem dfnul3 3497

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		equid 1749	. . . . 5 |- x = x
2	1	notnoti 650	. . . 4 |- -. -. x = x
3		pm3.24 700	. . . 4 |- -. ( x e. A /\ -. x e. A )
4	2, 3	2false 708	. . 3 |- ( -. x = x <-> ( x e. A /\ -. x e. A ) )
5	4	abbii 2347	. 2 |- { x | -. x = x } = { x | ( x e. A /\ -. x e. A ) }
6		dfnul2 3496	. 2 |- (/) = { x | -. x = x }
7		df-rab 2519	. 2 |- { x e. A | -. x e. A } = { x | ( x e. A /\ -. x e. A ) }
8	5, 6, 7	3eqtr4i 2262	1 |- (/) = { x e. A | -. x e. A }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514   (/)c0 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-dif 3202  df-nul 3495

This theorem is referenced by:  difidALT  3564