Theorem elunii 3855

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 2269	. . . . 5 |- ( x = B -> ( A e. x <-> A e. B ) )
2		eleq1 2268	. . . . 5 |- ( x = B -> ( x e. C <-> B e. C ) )
3	1, 2	anbi12d 473	. . . 4 |- ( x = B -> ( ( A e. x /\ x e. C ) <-> ( A e. B /\ B e. C ) ) )
4	3	spcegv 2861	. . 3 |- ( B e. C -> ( ( A e. B /\ B e. C ) -> E. x ( A e. x /\ x e. C ) ) )
5	4	anabsi7 581	. 2 |- ( ( A e. B /\ B e. C ) -> E. x ( A e. x /\ x e. C ) )
6		eluni 3853	. 2 |- ( A e. U. C <-> E. x ( A e. x /\ x e. C ) )
7	5, 6	sylibr 134	1 |- ( ( A e. B /\ B e. C ) -> A e. U. C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   E.wex 1515   e. wcel 2176   U.cuni 3850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-uni 3851

This theorem is referenced by:  ssuni  3872  unipw  4261  opeluu  4497  sucunielr  4558  unon  4559  ordunisuc2r  4562  tfrlemibxssdm  6413  tfr1onlemsucaccv  6427  tfr1onlembxssdm  6429  tfrcllemsucaccv  6440  tfrcllembxssdm  6442  wrdexb  11006  tgss2  14551  neipsm  14626  unirnblps  14894  unirnbl  14895  blbas  14905