Theorem disj1 2308

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 groups:   x,A   x,B

Proof of Theorem disj1

Step	Hyp	Ref	Expression
1		disj 2307	. 2 |- ((A i^i B) = (/) <-> A.x e. A -. x e. B)
2		df-ral 1646	. 2 |- (A.x e. A -. x e. B <-> A.x(x e. A -> -. x e. B))
3	1, 2	bitr 173	1 |- ((A i^i B) = (/) <-> A.x(x e. A -> -. x e. B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146  A.wal 952   = wceq 954   e. wcel 956  A.wral 1642   i^i cin 2042  (/)c0 2276

This theorem is referenced by:  reldisj 2309  disj3 2310  undif4 2321  disjssun 2322  disjsn 2437  funun 3546  erdisj 4276  aceq5lem4 4718  cnfilca 10487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-dif 2045  df-in 2047  df-nul 2277

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