Theorem ssdifeq0 3476

Description: A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)

Assertion

Ref	Expression
ssdifeq0	|- ( A C_ ( B \ A ) <-> A = (/) )

Proof of Theorem ssdifeq0

Step	Hyp	Ref	Expression
1		inidm 3316	. . 3 |- ( A i^i A ) = A
2		ssdifin0 3475	. . 3 |- ( A C_ ( B \ A ) -> ( A i^i A ) = (/) )
3	1, 2	eqtr3id 2204	. 2 |- ( A C_ ( B \ A ) -> A = (/) )
4		0ss 3432	. . 3 |- (/) C_ ( B \ (/) )
5		id 19	. . . 4 |- ( A = (/) -> A = (/) )
6		difeq2 3219	. . . 4 |- ( A = (/) -> ( B \ A ) = ( B \ (/) ) )
7	5, 6	sseq12d 3159	. . 3 |- ( A = (/) -> ( A C_ ( B \ A ) <-> (/) C_ ( B \ (/) ) ) )
8	4, 7	mpbiri 167	. 2 |- ( A = (/) -> A C_ ( B \ A ) )
9	3, 8	impbii 125	1 |- ( A C_ ( B \ A ) <-> A = (/) )

Syntax hints:   <-> wb 104   = wceq 1335   \ cdif 3099   i^i cin 3101   C_ wss 3102   (/)c0 3394

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395

This theorem is referenced by: (None)