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Theorem pwuniss 29708
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 4595 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴 𝒫 𝐵)
2 unipw 5046 . 2 𝒫 𝐵 = 𝐵
31, 2syl6sseq 3800 1 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3723  𝒫 cpw 4297   cuni 4574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-pr 4319  df-uni 4575
This theorem is referenced by:  elpwunicl  29709  pwldsys  30560
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