Theorem gruiin 10848

Description: A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
gruiin	⊢ ((𝑈 ∈ Univ ∧ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)

Distinct variable groups:   𝑥,𝑈   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem gruiin

Step	Hyp	Ref	Expression
1		nfv 1912	. . 3 ⊢ Ⅎ𝑥 𝑈 ∈ Univ
2		nfii1 5034	. . . 4 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵
3	2	nfel1 2920	. . 3 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈
4		iinss2 5062	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)
5		gruss 10834	. . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
6	4, 5	syl3an3 1164	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ 𝑥 ∈ 𝐴) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
7	6	3exp 1118	. . . 4 ⊢ (𝑈 ∈ Univ → (𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)))
8	7	com23 86	. . 3 ⊢ (𝑈 ∈ Univ → (𝑥 ∈ 𝐴 → (𝐵 ∈ 𝑈 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)))
9	1, 3, 8	rexlimd 3264	. 2 ⊢ (𝑈 ∈ Univ → (∃𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈))
10	9	imp 406	1 ⊢ ((𝑈 ∈ Univ ∧ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  ∃wrex 3068   ⊆ wss 3963  ∩ ciin 4997  Univcgru 10828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iin 4999  df-br 5149  df-tr 5266  df-iota 6516  df-fv 6571  df-ov 7434  df-gru 10829

This theorem is referenced by: (None)