Theorem ssdifeq0 4051

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 3822	. . 3 |- ( A i^i A ) = A
2		ssdifin0 4050	. . 3 |- ( A C_ ( B \ A ) -> ( A i^i A ) = (/) )
3	1, 2	syl5eqr 2670	. 2 |- ( A C_ ( B \ A ) -> A = (/) )
4		0ss 3972	. . 3 |- (/) C_ ( B \ (/) )
5		id 22	. . . 4 |- ( A = (/) -> A = (/) )
6		difeq2 3722	. . . 4 |- ( A = (/) -> ( B \ A ) = ( B \ (/) ) )
7	5, 6	sseq12d 3634	. . 3 |- ( A = (/) -> ( A C_ ( B \ A ) <-> (/) C_ ( B \ (/) ) ) )
8	4, 7	mpbiri 248	. 2 |- ( A = (/) -> A C_ ( B \ A ) )
9	3, 8	impbii 199	1 |- ( A C_ ( B \ A ) <-> A = (/) )

Syntax hints:   <-> wb 196   = wceq 1483   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  disjdifprg  29388  measxun2  30273  measssd  30278  pmeasmono  30386