Theorem elunii 3548

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 2074	. . . . 5 |- (x = B -> (A e. x <-> A e. B))
2		eleq1 2073	. . . . 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 2609	. . 3 |- (B e. C -> ((A e. B /\ B e. C) -> E.x(A e. x /\ x e. C)))
5	4	anabsi7 500	. 2 |- ((A e. B /\ B e. C) -> E.x(A e. x /\ x e. C))
6		eluni 3546	. 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 1223  E.wex 1354   e. wcel 1366   U.cuni 3543

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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995

This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-uni 3544

This theorem is referenced by:  ssuni  3565  unipw  3916  opeluu  4120  sucunielr  4173  unon  4174  ordunisuc2r  4177  tfrlemibxssdm  5850