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Theorem gruiin 9511
 Description: A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruiin ((𝑈 ∈ Univ ∧ ∃𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
Distinct variable groups:   𝑥,𝑈   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem gruiin
StepHypRef Expression
1 nfv 1830 . . 3 𝑥 𝑈 ∈ Univ
2 nfii1 4487 . . . 4 𝑥 𝑥𝐴 𝐵
32nfel1 2765 . . 3 𝑥 𝑥𝐴 𝐵𝑈
4 iinss2 4508 . . . . . 6 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
5 gruss 9497 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝐵𝑈 𝑥𝐴 𝐵𝐵) → 𝑥𝐴 𝐵𝑈)
64, 5syl3an3 1353 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐵𝑈𝑥𝐴) → 𝑥𝐴 𝐵𝑈)
763exp 1256 . . . 4 (𝑈 ∈ Univ → (𝐵𝑈 → (𝑥𝐴 𝑥𝐴 𝐵𝑈)))
87com23 84 . . 3 (𝑈 ∈ Univ → (𝑥𝐴 → (𝐵𝑈 𝑥𝐴 𝐵𝑈)))
91, 3, 8rexlimd 3008 . 2 (𝑈 ∈ Univ → (∃𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵𝑈))
109imp 444 1 ((𝑈 ∈ Univ ∧ ∃𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540  ∩ ciin 4456  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  ax-sep 4709 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-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iin 4458  df-br 4584  df-tr 4681  df-iota 5768  df-fv 5812  df-ov 6552  df-gru 9492 This theorem is referenced by: (None)
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