Theorem ssdifeq0 3450

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 3290	. . 3 |- ( A i^i A ) = A
2		ssdifin0 3449	. . 3 |- ( A C_ ( B \ A ) -> ( A i^i A ) = (/) )
3	1, 2	syl5eqr 2187	. 2 |- ( A C_ ( B \ A ) -> A = (/) )
4		0ss 3406	. . 3 |- (/) C_ ( B \ (/) )
5		id 19	. . . 4 |- ( A = (/) -> A = (/) )
6		difeq2 3193	. . . 4 |- ( A = (/) -> ( B \ A ) = ( B \ (/) ) )
7	5, 6	sseq12d 3133	. . 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 1332   \ cdif 3073   i^i cin 3075   C_ wss 3076   (/)c0 3368

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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369

This theorem is referenced by: (None)