Theorem ssdifeq0 3305

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 3146	. . 3 |- ( A i^i A ) = A
2		ssdifin0 3304	. . 3 |- ( A C_ ( B \ A ) -> ( A i^i A ) = (/) )
3	1, 2	syl5eqr 2086	. 2 |- ( A C_ ( B \ A ) -> A = (/) )
4		0ss 3255	. . 3 |- (/) C_ ( B \ (/) )
5		id 19	. . . 4 |- ( A = (/) -> A = (/) )
6		difeq2 3056	. . . 4 |- ( A = (/) -> ( B \ A ) = ( B \ (/) ) )
7	5, 6	sseq12d 2974	. . 3 |- ( A = (/) -> ( A C_ ( B \ A ) <-> (/) C_ ( B \ (/) ) ) )
8	4, 7	mpbiri 157	. 2 |- ( A = (/) -> A C_ ( B \ A ) )
9	3, 8	impbii 117	1 |- ( A C_ ( B \ A ) <-> A = (/) )

Syntax hints:   <-> wb 98   = wceq 1243   \ cdif 2914   i^i cin 2916   C_ wss 2917   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225

This theorem is referenced by: (None)