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

Theorem iunss2 4375
 Description: A subclass condition on the members of two indexed classes (x) and ( ) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 4282. (Contributed by NM, 9-Dec-2004.)
Assertion
Ref Expression
iunss2
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem iunss2
StepHypRef Expression
1 ssiun 4372 . . 3
21ralimi 2850 . 2
3 iunss 4371 . 2
42, 3sylibr 212 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  A.wral 2807  E.wrex 2808  C_wss 3475  U_ciun 4330 This theorem is referenced by:  iunxdif2  4378  oaass  7229  odi  7247  omass  7248  oelim2  7263 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-iun 4332
 Copyright terms: Public domain W3C validator