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Theorem gruiin 10209
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 1916 . . 3 𝑥 𝑈 ∈ Univ
2 nfii1 4927 . . . 4 𝑥 𝑥𝐴 𝐵
32nfel1 2990 . . 3 𝑥 𝑥𝐴 𝐵𝑈
4 iinss2 4954 . . . . . 6 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
5 gruss 10195 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝐵𝑈 𝑥𝐴 𝐵𝐵) → 𝑥𝐴 𝐵𝑈)
64, 5syl3an3 1162 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐵𝑈𝑥𝐴) → 𝑥𝐴 𝐵𝑈)
763exp 1116 . . . 4 (𝑈 ∈ Univ → (𝐵𝑈 → (𝑥𝐴 𝑥𝐴 𝐵𝑈)))
87com23 86 . . 3 (𝑈 ∈ Univ → (𝑥𝐴 → (𝐵𝑈 𝑥𝐴 𝐵𝑈)))
91, 3, 8rexlimd 3303 . 2 (𝑈 ∈ Univ → (∃𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵𝑈))
109imp 410 1 ((𝑈 ∈ Univ ∧ ∃𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wrex 3127  wss 3910   ciin 4893  Univcgru 10189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iin 4895  df-br 5040  df-tr 5146  df-iota 6287  df-fv 6336  df-ov 7133  df-gru 10190
This theorem is referenced by: (None)
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