Theorem ssdifeq0 3489

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 3329	. . 3 |- ( A i^i A ) = A
2		ssdifin0 3488	. . 3 |- ( A C_ ( B \ A ) -> ( A i^i A ) = (/) )
3	1, 2	eqtr3id 2211	. 2 |- ( A C_ ( B \ A ) -> A = (/) )
4		0ss 3445	. . 3 |- (/) C_ ( B \ (/) )
5		id 19	. . . 4 |- ( A = (/) -> A = (/) )
6		difeq2 3232	. . . 4 |- ( A = (/) -> ( B \ A ) = ( B \ (/) ) )
7	5, 6	sseq12d 3171	. . 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 1342   \ cdif 3111   i^i cin 3113   C_ wss 3114   (/)c0 3407

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rab 2451  df-v 2726  df-dif 3116  df-in 3120  df-ss 3127  df-nul 3408

This theorem is referenced by: (None)