Theorem elunii 3606

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 2142	. . . . 5 |- ( x = B -> ( A e. x <-> A e. B ) )
2		eleq1 2141	. . . . 5 |- ( x = B -> ( x e. C <-> B e. C ) )
3	1, 2	anbi12d 456	. . . 4 |- ( x = B -> ( ( A e. x /\ x e. C ) <-> ( A e. B /\ B e. C ) ) )
4	3	spcegv 2686	. . 3 |- ( B e. C -> ( ( A e. B /\ B e. C ) -> E. x ( A e. x /\ x e. C ) ) )
5	4	anabsi7 545	. 2 |- ( ( A e. B /\ B e. C ) -> E. x ( A e. x /\ x e. C ) )
6		eluni 3604	. 2 |- ( A e. U. C <-> E. x ( A e. x /\ x e. C ) )
7	5, 6	sylibr 132	1 |- ( ( A e. B /\ B e. C ) -> A e. U. C )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-uni 3602

This theorem is referenced by:  ssuni  3623  unipw  3972  opeluu  4200  sucunielr  4254  unon  4255  ordunisuc2r  4258  tfrlemibxssdm  5964