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

Theorem ssuni 4271
 Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
ssuni

Proof of Theorem ssuni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2530 . . . . . . 7
21imbi1d 317 . . . . . 6
3 elunii 4254 . . . . . . 7
43expcom 435 . . . . . 6
52, 4vtoclga 3173 . . . . 5
65imim2d 52 . . . 4
76alimdv 1709 . . 3
8 dfss2 3492 . . 3
9 dfss2 3492 . . 3
107, 8, 93imtr4g 270 . 2
1110impcom 430 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  C_wss 3475  U.cuni 4249 This theorem is referenced by:  elssuni  4279  uniss2  4282  ssorduni  6621  filssufilg  20412  alexsubALTlem2  20548  utoptop  20737  locfinreflem  27843 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-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-v 3111  df-in 3482  df-ss 3489  df-uni 4250
 Copyright terms: Public domain W3C validator