Theorem dfnul4 3522

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-13 2211, df-clel 2234. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 3523. (Revised by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul4	|- (/) = { x | F. }

Proof of Theorem dfnul4

Step	Hyp	Ref	Expression
1		df-nul 3521	. 2 |- (/) = ( _V \ _V )
2		df-dif 3222	. 2 |- ( _V \ _V ) = { x | ( x e. _V /\ -. x e. _V ) }
3		pm3.24 705	. . . 4 |- -. ( x e. _V /\ -. x e. _V )
4	3	bifal 1415	. . 3 |- ( ( x e. _V /\ -. x e. _V ) <-> F. )
5	4	abbii 2354	. 2 |- { x | ( x e. _V /\ -. x e. _V ) } = { x | F. }
6	1, 2, 5	3eqtri 2263	1 |- (/) = { x | F. }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1402   F. wfal 1407   e. wcel 2209   {cab 2224   _Vcvv 2821   \ cdif 3217   (/)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-dif 3222  df-nul 3521

This theorem is referenced by:  dfnul2  3523  dfnul3  3524  ab0w  3550