Theorem clel3g 1895

Description: An alternate definition of class membership when the class is a set.

Assertion

Ref	Expression
clel3g	|- (B e. C -> (A e. B <-> E.x(x = B /\ A e. x)))

Distinct variable groups:   x,A   x,B

Proof of Theorem clel3g

Step	Hyp	Ref	Expression
1		eleq2 1538	. . 3 |- (x = B -> (A e. x <-> A e. B))
2	1	ceqsexgv 1891	. 2 |- (B e. C -> (E.x(x = B /\ A e. x) <-> A e. B))
3	2	bicomd 523	1 |- (B e. C -> (A e. B <-> E.x(x = B /\ A e. x)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982

This theorem is referenced by:  clel3 1896  dfiun2g 2590

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

Copyright terms: Public domain