Theorem difidALT 3593

Description: The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3592. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
difidALT	|- ( A \ A ) = (/)

Proof of Theorem difidALT

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdif2 3228	. 2 |- ( A \ A ) = { x e. A | -. x e. A }
2		dfnul3 3524	. 2 |- (/) = { x e. A | -. x e. A }
3	1, 2	eqtr4i 2262	1 |- ( A \ A ) = (/)

Syntax hints:   -. wn 3   = wceq 1402   e. wcel 2209   {crab 2532   \ 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-rab 2537  df-dif 3222  df-nul 3521

This theorem is referenced by: (None)