Theorem ssindif0 1741

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 1735	. 2 |- ((A i^i (V \ B)) = (/) <-> A (_ (V \ (V \ B)))
2		ddif 1597	. . 3 |- (V \ (V \ B)) = B
3	2	sseq2i 1525	. 2 |- (A (_ (V \ (V \ B)) <-> A (_ B)
4	1, 3	bitr2 152	1 |- (A (_ B <-> (A i^i (V \ B)) = (/))

Syntax hints:   <-> wb 127   = wceq 1091  Vcvv 1348   \ cdif 1484   i^i cin 1486   (_ wss 1487  (/)c0 1707

This theorem is referenced by:  setind 3492

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708

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