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Theorem pwuniss 29213
Description: Condition for a class union to be a subset. (Contributed by Thierry Arnoux, 21-Jun-2020.)
Assertion
Ref Expression
pwuniss (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)

Proof of Theorem pwuniss
StepHypRef Expression
1 uniss 4424 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴 𝒫 𝐵)
2 unipw 4879 . 2 𝒫 𝐵 = 𝐵
31, 2syl6sseq 3630 1 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  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  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  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-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132  df-sn 4149  df-pr 4151  df-uni 4403
This theorem is referenced by:  elpwunicl  29214  pwldsys  29998
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