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Theorem gruss 9603
Description: Any subset of an element of a Grothendieck universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruss ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝐴) → 𝐵𝑈)

Proof of Theorem gruss
StepHypRef Expression
1 elpw2g 4818 . . . 4 (𝐴𝑈 → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
21adantl 482 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
3 grupw 9602 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝒫 𝐴𝑈)
4 gruelss 9601 . . . . 5 ((𝑈 ∈ Univ ∧ 𝒫 𝐴𝑈) → 𝒫 𝐴𝑈)
53, 4syldan 487 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝒫 𝐴𝑈)
65sseld 3594 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐵 ∈ 𝒫 𝐴𝐵𝑈))
72, 6sylbird 250 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐵𝐴𝐵𝑈))
873impia 1259 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝐴) → 𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1988  wss 3567  𝒫 cpw 4149  Univcgru 9597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-tr 4744  df-iota 5839  df-fv 5884  df-ov 6638  df-gru 9598
This theorem is referenced by:  grurn  9608  gruima  9609  gruxp  9614  grumap  9615  gruixp  9616  gruiin  9617  grudomon  9624  gruina  9625
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