Theorem ssdifeq0 3497

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 3336	. . 3 |- ( A i^i A ) = A
2		ssdifin0 3496	. . 3 |- ( A C_ ( B \ A ) -> ( A i^i A ) = (/) )
3	1, 2	eqtr3id 2217	. 2 |- ( A C_ ( B \ A ) -> A = (/) )
4		0ss 3453	. . 3 |- (/) C_ ( B \ (/) )
5		id 19	. . . 4 |- ( A = (/) -> A = (/) )
6		difeq2 3239	. . . 4 |- ( A = (/) -> ( B \ A ) = ( B \ (/) ) )
7	5, 6	sseq12d 3178	. . 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 1348   \ cdif 3118   i^i cin 3120   C_ wss 3121   (/)c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by: (None)