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

Theorem elpwunicl 30234
Description: Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.)
Hypotheses
Ref Expression
elpwunicl.1 (𝜑𝐵𝑉)
elpwunicl.2 (𝜑𝐴 ∈ 𝒫 𝒫 𝐵)
Assertion
Ref Expression
elpwunicl (𝜑 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwunicl
StepHypRef Expression
1 elpwunicl.2 . . . 4 (𝜑𝐴 ∈ 𝒫 𝒫 𝐵)
2 elpwg 4541 . . . . 5 (𝐴 ∈ 𝒫 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝒫 𝐵𝐴 ⊆ 𝒫 𝐵))
31, 2syl 17 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝒫 𝐵𝐴 ⊆ 𝒫 𝐵))
41, 3mpbid 233 . . 3 (𝜑𝐴 ⊆ 𝒫 𝐵)
5 pwuniss 30233 . . 3 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
64, 5syl 17 . 2 (𝜑 𝐴𝐵)
7 uniexg 7456 . . 3 (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ V)
8 elpwg 4541 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐵 𝐴𝐵))
91, 7, 83syl 18 . 2 (𝜑 → ( 𝐴 ∈ 𝒫 𝐵 𝐴𝐵))
106, 9mpbird 258 1 (𝜑 𝐴 ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wcel 2105  Vcvv 3492  wss 3933  𝒫 cpw 4535   cuni 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-pw 4537  df-sn 4558  df-pr 4560  df-uni 4831
This theorem is referenced by:  ldgenpisyslem1  31321
  Copyright terms: Public domain W3C validator