Theorem ssindif0 2322

Description: Subclass expressed in terms of intersection with difference from the universal class.

Assertion

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

Proof of Theorem ssindif0

Step	Hyp	Ref	Expression
1		disj2 2316	. 2 |- ((A i^i (V \ B)) = (/) <-> A (_ (V \ (V \ B)))
2		ddif 2169	. . 3 |- (V \ (V \ B)) = B
3	2	sseq2i 2086	. 2 |- (A (_ (V \ (V \ B)) <-> A (_ B)
4	1, 3	bitr2 174	1 |- (A (_ B <-> (A i^i (V \ B)) = (/))

Syntax hints:   <-> wb 146   = wceq 956  Vcvv 1811   \ cdif 2044   i^i cin 2046   (_ wss 2047  (/)c0 2280

This theorem is referenced by:  setind 4648

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281

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