Theorem elunii 4096

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 2504	. . . . 5 |- ( x = B -> ( A e. x <-> A e. B ) )
2		eleq1 2503	. . . . 5 |- ( x = B -> ( x e. C <-> B e. C ) )
3	1, 2	anbi12d 710	. . . 4 |- ( x = B -> ( ( A e. x /\ x e. C ) <-> ( A e. B /\ B e. C ) ) )
4	3	spcegv 3058	. . 3 |- ( B e. C -> ( ( A e. B /\ B e. C ) -> E. x ( A e. x /\ x e. C ) ) )
5	4	anabsi7 815	. 2 |- ( ( A e. B /\ B e. C ) -> E. x ( A e. x /\ x e. C ) )
6		eluni 4094	. 2 |- ( A e. U. C <-> E. x ( A e. x /\ x e. C ) )
7	5, 6	sylibr 212	1 |- ( ( A e. B /\ B e. C ) -> A e. U. C )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   E.wex 1586   e. wcel 1756   U.cuni 4091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2974  df-uni 4092

This theorem is referenced by:  ssuni  4113  unipw  4542  opeluu  4561  unon  6442  limuni3  6463  uniinqs  7180  trcl  7948  rankwflemb  8000  ac5num  8206  dfac3  8291  isf34lem4  8546  axcclem  8626  ttukeylem7  8684  brdom7disj  8698  brdom6disj  8699  wrdexb  12245  dprdfeq0  16512  dprdfeq0OLD  16519  tgss2  18592  ppttop  18611  isclo  18691  neips  18717  bwthOLD  19014  2ndcomap  19062  2ndcsep  19063  txkgen  19225  txcon  19262  basqtop  19284  nrmr0reg  19322  alexsublem  19616  alexsubALTlem4  19622  alexsubALT  19623  ptcmplem4  19627  unirnblps  19994  unirnbl  19995  blbas  20005  met2ndci  20097  bndth  20530  dyadmbllem  21079  opnmbllem  21081  dya2iocnei  26697  dstfrvunirn  26857  pconcon  27120  cvmcov2  27164  cvmlift2lem11  27202  cvmlift2lem12  27203  onint1  28295  opnmbllem0  28427  locfincmp  28576  comppfsc  28579  neibastop2lem  28581  heibor1  28709  unichnidl  28831  prtlem16  29014  prter2  29026  stoweidlem43  29838  stoweidlem55  29850  truniALT  31248  unipwrVD  31568  unipwr  31569  truniALTVD  31614  unisnALT  31662