Theorem ssdifeq0 3633

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 3465	. . 3 |- (A i^i A) = A
2		ssdifin0 3632	. . 3 |- (A C_ (B \ A) -> (A i^i A) = (/))
3	1, 2	syl5eqr 2399	. 2 |- (A C_ (B \ A) -> A = (/))
4		0ss 3580	. . 3 |- (/) C_ (B \ (/))
5		id 19	. . . 4 |- (A = (/) -> A = (/))
6		difeq2 3248	. . . 4 |- (A = (/) -> (B \ A) = (B \ (/)))
7	5, 6	sseq12d 3301	. . 3 |- (A = (/) -> (A C_ (B \ A) <-> (/) C_ (B \ (/))))
8	4, 7	mpbiri 224	. 2 |- (A = (/) -> A C_ (B \ A))
9	3, 8	impbii 180	1 |- (A C_ (B \ A) <-> A = (/))

Syntax hints:   <-> wb 176   = wceq 1642   \ cdif 3207   i^i cin 3209   C_ wss 3258  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)