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Theorem gruiin 9576
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 1840 . . 3 𝑥 𝑈 ∈ Univ
2 nfii1 4517 . . . 4 𝑥 𝑥𝐴 𝐵
32nfel1 2775 . . 3 𝑥 𝑥𝐴 𝐵𝑈
4 iinss2 4538 . . . . . 6 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
5 gruss 9562 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝐵𝑈 𝑥𝐴 𝐵𝐵) → 𝑥𝐴 𝐵𝑈)
64, 5syl3an3 1358 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐵𝑈𝑥𝐴) → 𝑥𝐴 𝐵𝑈)
763exp 1261 . . . 4 (𝑈 ∈ Univ → (𝐵𝑈 → (𝑥𝐴 𝑥𝐴 𝐵𝑈)))
87com23 86 . . 3 (𝑈 ∈ Univ → (𝑥𝐴 → (𝐵𝑈 𝑥𝐴 𝐵𝑈)))
91, 3, 8rexlimd 3019 . 2 (𝑈 ∈ Univ → (∃𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵𝑈))
109imp 445 1 ((𝑈 ∈ Univ ∧ ∃𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wrex 2908  wss 3555   ciin 4486  Univcgru 9556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iin 4488  df-br 4614  df-tr 4713  df-iota 5810  df-fv 5855  df-ov 6607  df-gru 9557
This theorem is referenced by: (None)
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