Theorem elunii 2508

Description: Membership in class union.

Assertion

Ref	Expression
elunii	|- ((A e. B /\ B e. C) -> A e. U.C)

Proof of Theorem elunii

Step	Hyp	Ref	Expression
1		eleq2 1535	. . . . 5 |- (x = B -> (A e. x <-> A e. B))
2		eleq1 1534	. . . . 5 |- (x = B -> (x e. C <-> B e. C))
3	1, 2	anbi12d 628	. . . 4 |- (x = B -> ((A e. x /\ x e. C) <-> (A e. B /\ B e. C)))
4	3	cla4egv 1863	. . 3 |- (B e. C -> ((A e. B /\ B e. C) -> E.x(A e. x /\ x e. C)))
5	4	anabsi7 497	. 2 |- ((A e. B /\ B e. C) -> E.x(A e. x /\ x e. C))
6		eluni 2506	. 2 |- (A e. U.C <-> E.x(A e. x /\ x e. C))
7	5, 6	sylibr 200	1 |- ((A e. B /\ B e. C) -> A e. U.C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  U.cuni 2503

This theorem is referenced by:  opeluu 2879  unon 3088  limuni3 3123  trcl 4645  aceq3 4733  brdom7disj 4804  brdom6disj 4805  suplem1pr 5161  neips 7727

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-uni 2504

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