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

Theorem pwuncl 7789
Description: Power classes are closed under union. (Contributed by AV, 27-Feb-2024.)
Assertion
Ref Expression
pwuncl ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ 𝒫 𝑋)

Proof of Theorem pwuncl
StepHypRef Expression
1 unexg 7762 . 2 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ V)
2 elpwi 4612 . . 3 (𝐴 ∈ 𝒫 𝑋𝐴𝑋)
3 elpwi 4612 . . 3 (𝐵 ∈ 𝒫 𝑋𝐵𝑋)
4 unss 4200 . . . 4 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
54biimpi 216 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
62, 3, 5syl2an 596 . 2 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ⊆ 𝑋)
71, 6elpwd 4611 1 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  Vcvv 3478  cun 3961  wss 3963  𝒫 cpw 4605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913
This theorem is referenced by:  naddunif  8730  fiin  9460  fpwipodrs  18598  pwmnd  18963  cutlt  27981  clsk1indlem3  44033  isotone1  44038  isgrtri  47848
  Copyright terms: Public domain W3C validator