Theorem difidALT 3538

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 3537. (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 3182	. 2 |- ( A \ A ) = { x e. A | -. x e. A }
2		dfnul3 3471	. 2 |- (/) = { x e. A | -. x e. A }
3	1, 2	eqtr4i 2231	1 |- ( A \ A ) = (/)

Syntax hints:   -. wn 3   = wceq 1373   e. wcel 2178   {crab 2490   \ cdif 3171   (/)c0 3468

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778  df-dif 3176  df-nul 3469

This theorem is referenced by: (None)