Theorem elunii 2512

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 1538	. . . . 5 |- (x = B -> (A e. x <-> A e. B))
2		eleq1 1537	. . . . 5 |- (x = B -> (x e. C <-> B e. C))
3	1, 2	anbi12d 630	. . . 4 |- (x = B -> ((A e. x /\ x e. C) <-> (A e. B /\ B e. C)))
4	3	cla4egv 1866	. . 3 |- (B e. C -> ((A e. B /\ B e. C) -> E.x(A e. x /\ x e. C)))
5	4	anabsi7 499	. 2 |- ((A e. B /\ B e. C) -> E.x(A e. x /\ x e. C))
6		eluni 2510	. 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 958   e. wcel 960  E.wex 982  U.cuni 2507

This theorem is referenced by:  opeluu 2885  unon 3094  limuni3 3129  trcl 4655  aceq3 4743  brdom7disj 4814  brdom6disj 4815  suplem1pr 5173  neips 7724

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-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  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  df-uni 2508

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