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Theorem wunss 9478
Description: A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
wunss.3 (𝜑𝐵𝐴)
Assertion
Ref Expression
wunss (𝜑𝐵𝑈)

Proof of Theorem wunss
StepHypRef Expression
1 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
2 wununi.2 . . . 4 (𝜑𝐴𝑈)
31, 2wunpw 9473 . . 3 (𝜑 → 𝒫 𝐴𝑈)
41, 3wunelss 9474 . 2 (𝜑 → 𝒫 𝐴𝑈)
5 wunss.3 . . 3 (𝜑𝐵𝐴)
62, 5sselpwd 4767 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
74, 6sseldd 3584 1 (𝜑𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wss 3555  𝒫 cpw 4130  WUnicwun 9466
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
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-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132  df-uni 4403  df-tr 4713  df-wun 9468
This theorem is referenced by:  wunin  9479  wundif  9480  wunint  9481  wun0  9484  wunom  9486  wunxp  9490  wunpm  9491  wunmap  9492  wundm  9494  wunrn  9495  wuncnv  9496  wunres  9497  wunfv  9498  wunco  9499  wuntpos  9500  wuncn  9935  wunndx  15800  wunstr  15803  wunfunc  16480
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