Theorem elunii 3585

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 2101	. . . . 5 |- ( x = B -> ( A e. x <-> A e. B ) )
2		eleq1 2100	. . . . 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 2641	. . 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 3583	. 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 1243   E.wex 1381   e. wcel 1393   U.cuni 3580

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-uni 3581

This theorem is referenced by:  ssuni  3602  unipw  3953  opeluu  4182  sucunielr  4236  unon  4237  ordunisuc2r  4240  tfrlemibxssdm  5941