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Theorem gruel 9504
Description: Any element of an element of a Grothendieck universe is also an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruel ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝐴) → 𝐵𝑈)

Proof of Theorem gruel
StepHypRef Expression
1 gruelss 9495 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
21sseld 3567 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐵𝐴𝐵𝑈))
323impia 1253 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝐴) → 𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031  wcel 1977  Univcgru 9491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-tr 4681  df-iota 5768  df-fv 5812  df-ov 6552  df-gru 9492
This theorem is referenced by:  gruf  9512
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