Theorem elint2 2536

Description: Membership in class intersection.

Hypothesis

Ref	Expression
elint2.1	|- A e. V

Assertion

Ref	Expression
elint2	|- (A e. |^|B <-> A.x e. B A e. x)

Distinct variable groups:   x,A   x,B

Proof of Theorem elint2

Step	Hyp	Ref	Expression
1		elint2.1	. . 3 |- A e. V
2	1	elint 2535	. 2 |- (A e. |^|B <-> A.x(x e. B -> A e. x))
3		df-ral 1647	. 2 |- (A.x e. B A e. x <-> A.x(x e. B -> A e. x))
4	2, 3	bitr4 176	1 |- (A e. |^|B <-> A.x e. B A e. x)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 953   e. wcel 957  A.wral 1643  Vcvv 1808  |^|cint 2529

This theorem is referenced by:  elintg 2537  ssint 2545  intssuni 2551  iinuni 2611  onint 3002  shintcl 9248  chintcl 9250

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-v 1809  df-int 2530

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