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Theorem gruiin 10594
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 1913 . . 3 𝑥 𝑈 ∈ Univ
2 nfii1 4962 . . . 4 𝑥 𝑥𝐴 𝐵
32nfel1 2918 . . 3 𝑥 𝑥𝐴 𝐵𝑈
4 iinss2 4990 . . . . . 6 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
5 gruss 10580 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝐵𝑈 𝑥𝐴 𝐵𝐵) → 𝑥𝐴 𝐵𝑈)
64, 5syl3an3 1163 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐵𝑈𝑥𝐴) → 𝑥𝐴 𝐵𝑈)
763exp 1117 . . . 4 (𝑈 ∈ Univ → (𝐵𝑈 → (𝑥𝐴 𝑥𝐴 𝐵𝑈)))
87com23 86 . . 3 (𝑈 ∈ Univ → (𝑥𝐴 → (𝐵𝑈 𝑥𝐴 𝐵𝑈)))
91, 3, 8rexlimd 3278 . 2 (𝑈 ∈ Univ → (∃𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵𝑈))
109imp 406 1 ((𝑈 ∈ Univ ∧ ∃𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2101  wrex 3068  wss 3889   ciin 4928  Univcgru 10574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iin 4930  df-br 5078  df-tr 5195  df-iota 6399  df-fv 6455  df-ov 7298  df-gru 10575
This theorem is referenced by: (None)
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