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Theorem pwpwssunieq 5107
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 4040 . . 3 ( 𝑥 = 𝐴 𝑥𝐴)
21ss2abi 4063 . 2 {𝑥 𝑥 = 𝐴} ⊆ {𝑥 𝑥𝐴}
3 pwpwab 5106 . 2 𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
42, 3sseqtrri 4019 1 {𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  {cab 2710  wss 3948  𝒫 cpw 4602   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3955  df-ss 3965  df-pw 4604  df-uni 4909
This theorem is referenced by:  toponsspwpw  22416  dmtopon  22417
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