Theorem dfnul3 3524

Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.)

Assertion

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

Proof of Theorem dfnul3

Step	Hyp	Ref	Expression
1		fal 1409	. . . 4 |- -. F.
2		pm3.24 705	. . . 4 |- -. ( x e. A /\ -. x e. A )
3	1, 2	2false 713	. . 3 |- ( F. <-> ( x e. A /\ -. x e. A ) )
4	3	abbii 2354	. 2 |- { x | F. } = { x | ( x e. A /\ -. x e. A ) }
5		dfnul4 3522	. 2 |- (/) = { x | F. }
6		df-rab 2537	. 2 |- { x e. A | -. x e. A } = { x | ( x e. A /\ -. x e. A ) }
7	4, 5, 6	3eqtr4i 2269	1 |- (/) = { x e. A | -. x e. A }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1402   F. wfal 1407   e. wcel 2209   {cab 2224   {crab 2532   (/)c0 3520

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 623  ax-in2 624  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-rab 2537  df-dif 3222  df-nul 3521

This theorem is referenced by:  difidALT  3593