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Theorem bj-sspwpweq 32240
Description: The class of families 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
bj-sspwpweq {𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-sspwpweq
StepHypRef Expression
1 eqimss 3620 . . 3 ( 𝑥 = 𝐴 𝑥𝐴)
21ss2abi 3637 . 2 {𝑥 𝑥 = 𝐴} ⊆ {𝑥 𝑥𝐴}
3 bj-sspwpwab 32239 . 2 {𝑥 𝑥𝐴} = 𝒫 𝒫 𝐴
42, 3sseqtri 3600 1 {𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  {cab 2596  wss 3540  𝒫 cpw 4108   cuni 4372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373
This theorem is referenced by:  bj-toponss  32241  bj-dmtopon  32242
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