MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unss2 Structured version   Visualization version   GIF version

Theorem unss2 4181
Description: Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
unss2 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Proof of Theorem unss2
StepHypRef Expression
1 unss1 4179 . 2 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 uncom 4153 . 2 (𝐶𝐴) = (𝐴𝐶)
3 uncom 4153 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33sstr4g 4027 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cun 3946  wss 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3953  df-in 3955  df-ss 3965
This theorem is referenced by:  unss12  4182  ord3ex  5385  xpider  8784  fin1a2lem13  10409  canthp1lem2  10650  seqexw  13984  uniioombllem3  25109  volcn  25130  dvres2lem  25434  mulsproplem13  27594  mulsproplem14  27595  bnj1413  34115  bnj1408  34116
  Copyright terms: Public domain W3C validator