Theorem disj1 1734

Description: Two ways of saying that two classes are disjoint (have no members in common).

Assertion

Ref	Expression
disj1	|- ((A i^i B) = (/) <-> A.x(x e. A -> -. x e. B))

Distinct variable group(s):   x,A   x,B

Proof of Theorem disj1

Step	Hyp	Ref	Expression
1		disj 1733	. 2 |- ((A i^i B) = (/) <-> A.x e. A -. x e. B)
2		df-ral 1205	. . 3 |- (A.x e. A -. x e. B <-> A.x(x e. A -> -. x e. B))
3	2	bicomi 150	. 2 |- (A.x(x e. A -> -. x e. B) <-> A.x e. A -. x e. B)
4	1, 3	bitr4 154	1 |- ((A i^i B) = (/) <-> A.x(x e. A -> -. x e. B))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127  A.wal 672   = wceq 1091   e. wcel 1092  A.wral 1201   i^i cin 1486  (/)c0 1707

This theorem is referenced by:  disj2 1735  disj3 1736  undif4 1744  disjsn 1836  funun 2700  erdisj 3223  aceq5lem4 3561

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-nul 1708

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