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Theorem pwpwssunieq 3937
 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 3182 . . 3 ( 𝑥 = 𝐴 𝑥𝐴)
21ss2abi 3200 . 2 {𝑥 𝑥 = 𝐴} ⊆ {𝑥 𝑥𝐴}
3 pwpwab 3936 . 2 𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
42, 3sseqtrri 3163 1 {𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
 Colors of variables: wff set class Syntax hints:   = wceq 1335  {cab 2143   ⊆ wss 3102  𝒫 cpw 3543  ∪ cuni 3772 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545  df-uni 3773 This theorem is referenced by:  toponsspwpwg  12380  dmtopon  12381
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