Theorem difidALT 3293

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 3292. (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 2926	. 2 |- ( A \ A ) = { x e. A | -. x e. A }
2		dfnul3 3227	. 2 |- (/) = { x e. A | -. x e. A }
3	1, 2	eqtr4i 2063	1 |- ( A \ A ) = (/)

Syntax hints:   -. wn 3   = wceq 1243   e. wcel 1393   {crab 2310   \ cdif 2914   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-dif 2920  df-nul 3225

This theorem is referenced by: (None)