MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunss2 Unicode version

Theorem iunss2 4332
Description: A subclass condition on the members of two indexed classes (x) and (y) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 4241. (Contributed by NM, 9-Dec-2004.)
Assertion
Ref Expression
iunss2
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem iunss2
StepHypRef Expression
1 ssiun 4329 . . 3
21ralimi 2820 . 2
3 iunss 4328 . 2
42, 3sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  A.wral 2800  E.wrex 2801  C_wss 3442  U_ciun 4288
This theorem is referenced by:  iunxdif2  4335  oaass  7134  odi  7152  omass  7153  oelim2  7168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2805  df-rex 2806  df-v 3083  df-in 3449  df-ss 3456  df-iun 4290
  Copyright terms: Public domain W3C validator