MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unss1 Structured version   Visualization version   GIF version

Theorem unss1 4180
Description: Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
unss1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem unss1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel 3976 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21orim1d 965 . . 3 (𝐴𝐵 → ((𝑥𝐴𝑥𝐶) → (𝑥𝐵𝑥𝐶)))
3 elun 4149 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elun 4149 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
52, 3, 43imtr4g 296 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐴𝐶) → 𝑥 ∈ (𝐵𝐶)))
65ssrdv 3989 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846  wcel 2107  cun 3947  wss 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-ss 3966
This theorem is referenced by:  unss2  4182  unss12  4183  eldifpw  7755  orderseqlem  8143  tposss  8212  dftpos4  8230  hashbclem  14411  incexclem  15782  mreexexlem2d  17589  catcoppccl  18067  catcoppcclOLD  18068  neitr  22684  restntr  22686  leordtval2  22716  cmpcld  22906  uniioombllem3  25102  limcres  25403  plyss  25713  mulsproplem13  27584  mulsproplem14  27585  shlej1  30613  fineqvac  34097  ss2mcls  34559  bj-rrhatsscchat  36117  pclfinclN  38821
  Copyright terms: Public domain W3C validator