Theorem difidALT 3516

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 3515. (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 3161	. 2 |- ( A \ A ) = { x e. A | -. x e. A }
2		dfnul3 3449	. 2 |- (/) = { x e. A | -. x e. A }
3	1, 2	eqtr4i 2217	1 |- ( A \ A ) = (/)

Syntax hints:   -. wn 3   = wceq 1364   e. wcel 2164   {crab 2476   \ cdif 3150   (/)c0 3446

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-dif 3155  df-nul 3447

This theorem is referenced by: (None)