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Theorem bj-sspwpweq 32696
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 3636 . . 3 ( 𝑥 = 𝐴 𝑥𝐴)
21ss2abi 3653 . 2 {𝑥 𝑥 = 𝐴} ⊆ {𝑥 𝑥𝐴}
3 bj-sspwpwab 32695 . 2 {𝑥 𝑥𝐴} = 𝒫 𝒫 𝐴
42, 3sseqtri 3616 1 {𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  {cab 2607  wss 3555  𝒫 cpw 4130   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132  df-uni 4403
This theorem is referenced by:  bj-toponss  32697  bj-dmtopon  32698
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