Theorem difid 3458

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. (Contributed by NM, 22-Apr-2004.)

Assertion

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

Proof of Theorem difid

Step	Hyp	Ref	Expression
1		ssid 3144	. 2 |- A C_ A
2		ssdif0im 3454	. 2 |- ( A C_ A -> ( A \ A ) = (/) )
3	1, 2	ax-mp 5	1 |- ( A \ A ) = (/)

Syntax hints:   = wceq 1332   \ cdif 3095   C_ wss 3098   (/)c0 3390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100  df-in 3104  df-ss 3111  df-nul 3391

This theorem is referenced by:  dif0  3460  difun2  3469  diftpsn3  3693  2oconcl  6376  ismkvnex  7077  topcld  12448