Theorem difid 3493

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 3177	. 2 |- A C_ A
2		ssdif0im 3489	. 2 |- ( A C_ A -> ( A \ A ) = (/) )
3	1, 2	ax-mp 5	1 |- ( A \ A ) = (/)

Syntax hints:   = wceq 1353   \ cdif 3128   C_ wss 3131   (/)c0 3424

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425

This theorem is referenced by:  dif0  3495  difun2  3504  diftpsn3  3735  2oconcl  6442  ismkvnex  7155  topcld  13694