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Theorem pwpwssunieq 4989
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 3971 . . 3 ( 𝑥 = 𝐴 𝑥𝐴)
21ss2abi 3994 . 2 {𝑥 𝑥 = 𝐴} ⊆ {𝑥 𝑥𝐴}
3 pwpwab 4988 . 2 𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
42, 3sseqtrri 3952 1 {𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  {cab 2776  wss 3881  𝒫 cpw 4497   cuni 4800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499  df-uni 4801
This theorem is referenced by:  toponsspwpw  21527  dmtopon  21528
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