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

Theorem pwpwssunieq 5097
Description: The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
Assertion
Ref Expression
pwpwssunieq {𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem pwpwssunieq
StepHypRef Expression
1 eqimss 4032 . . 3 ( 𝑥 = 𝐴 𝑥𝐴)
21ss2abi 4055 . 2 {𝑥 𝑥 = 𝐴} ⊆ {𝑥 𝑥𝐴}
3 pwpwab 5096 . 2 𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
42, 3sseqtrri 4011 1 {𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  {cab 2701  wss 3940  𝒫 cpw 4594   cuni 4899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-v 3468  df-in 3947  df-ss 3957  df-pw 4596  df-uni 4900
This theorem is referenced by:  toponsspwpw  22746  dmtopon  22747
  Copyright terms: Public domain W3C validator