Theorem elunii 3576

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 2098	. . . . 5 |- (x = B -> (A e. x <-> A e. B))
2		eleq1 2097	. . . . 5 |- (x = B -> (x e. C <-> B e. C))
3	1, 2	anbi12d 442	. . . 4 |- (x = B -> ((A e. x /\ x e. C) <-> (A e. B /\ B e. C)))
4	3	spcegv 2635	. . 3 |- (B e. C -> ((A e. B /\ B e. C) -> E.x(A e. x /\ x e. C)))
5	4	anabsi7 515	. 2 |- ((A e. B /\ B e. C) -> E.x(A e. x /\ x e. C))
6		eluni 3574	. 2 |- (A e. U. C <-> E.x(A e. x /\ x e. C))
7	5, 6	sylibr 137	1 |- ((A e. B /\ B e. C) -> A e. U. C)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242  E.wex 1378   e. wcel 1390   U.cuni 3571

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-uni 3572

This theorem is referenced by:  ssuni  3593  unipw  3944  opeluu  4148  sucunielr  4201  unon  4202  ordunisuc2r  4205  tfrlemibxssdm  5882