Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwuniss Structured version   Visualization version   GIF version

Theorem pwuniss 28753
 Description: Condition for a class union to be a subset. (Contributed by Thierry Arnoux, 21-Jun-2020.)
Assertion
Ref Expression
pwuniss (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)

Proof of Theorem pwuniss
StepHypRef Expression
1 uniss 4394 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴 𝒫 𝐵)
2 unipw 4845 . 2 𝒫 𝐵 = 𝐵
31, 2syl6sseq 3614 1 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373 This theorem is referenced by:  elpwunicl  28754  pwldsys  29547
 Copyright terms: Public domain W3C validator