Theorem elunii 3869

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 2271	. . . . 5 |- ( x = B -> ( A e. x <-> A e. B ) )
2		eleq1 2270	. . . . 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 2868	. . 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 3867	. 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 1516   e. wcel 2178   U.cuni 3864

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-uni 3865

This theorem is referenced by:  ssuni  3886  unipw  4279  opeluu  4515  sucunielr  4576  unon  4577  ordunisuc2r  4580  tfrlemibxssdm  6436  tfr1onlemsucaccv  6450  tfr1onlembxssdm  6452  tfrcllemsucaccv  6463  tfrcllembxssdm  6465  wrdexb  11043  tgss2  14666  neipsm  14741  unirnblps  15009  unirnbl  15010  blbas  15020