Theorem ssindif0 4031

Description: Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Assertion

Ref	Expression
ssindif0	|- ( A C_ B <-> ( A i^i ( _V \ B ) ) = (/) )

Proof of Theorem ssindif0

Step	Hyp	Ref	Expression
1		disj2 4024	. 2 |- ( ( A i^i ( _V \ B ) ) = (/) <-> A C_ ( _V \ ( _V \ B ) ) )
2		ddif 3742	. . 3 |- ( _V \ ( _V \ B ) ) = B
3	2	sseq2i 3630	. 2 |- ( A C_ ( _V \ ( _V \ B ) ) <-> A C_ B )
4	1, 3	bitr2i 265	1 |- ( A C_ B <-> ( A i^i ( _V \ B ) ) = (/) )

Syntax hints:   <-> wb 196   = wceq 1483   _Vcvv 3200   \ 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-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  setind  8610