Theorem difid 3592

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 3268	. 2 |- A C_ A
2		ssdif0im 3588	. 2 |- ( A C_ A -> ( A \ A ) = (/) )
3	1, 2	ax-mp 5	1 |- ( A \ A ) = (/)

Syntax hints:   = wceq 1402   \ cdif 3217   C_ wss 3220   (/)c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by:  dif0  3594  difun2  3604  diftpsn3  3851  2oconcl  6702  ismkvnex  7485  topcld  15133