Theorem elunii 3897

Description: Membership in class union. (Contributed by NM, 24-Mar-1995.)

Assertion

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

Proof of Theorem elunii

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . . 5 |- (x = B -> (A e. x <-> A e. B))
2		eleq1 2413	. . . . 5 |- (x = B -> (x e. C <-> B e. C))
3	1, 2	anbi12d 691	. . . 4 |- (x = B -> ((A e. x /\ x e. C) <-> (A e. B /\ B e. C)))
4	3	spcegv 2941	. . 3 |- (B e. C -> ((A e. B /\ B e. C) -> E.x(A e. x /\ x e. C)))
5	4	anabsi7 792	. 2 |- ((A e. B /\ B e. C) -> E.x(A e. x /\ x e. C))
6		eluni 3895	. 2 |- (A e. U.C <-> E.x(A e. x /\ x e. C))
7	5, 6	sylibr 203	1 |- ((A e. B /\ B e. C) -> A e. U.C)

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  U.cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-uni 3893

This theorem is referenced by:  ssuni  3914  unipw  4118  nnadjoin  4521  sfinltfin  4536  nulnnn  4557