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

Theorem pwpwssunieq 4749
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 3806 . . 3 ( 𝑥 = 𝐴 𝑥𝐴)
21ss2abi 3823 . 2 {𝑥 𝑥 = 𝐴} ⊆ {𝑥 𝑥𝐴}
3 pwpwab 4748 . 2 𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
42, 3sseqtr4i 3787 1 {𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  {cab 2757  wss 3723  𝒫 cpw 4297   cuni 4574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-in 3730  df-ss 3737  df-pw 4299  df-uni 4575
This theorem is referenced by:  toponsspwpw  20947  dmtopon  20948
  Copyright terms: Public domain W3C validator