Theorem gruiin 10701

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 1915	. . 3 ⊢ Ⅎ𝑥 𝑈 ∈ Univ
2		nfii1 4977	. . . 4 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵
3	2	nfel1 2911	. . 3 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈
4		iinss2 5004	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)
5		gruss 10687	. . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
6	4, 5	syl3an3 1165	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ 𝑥 ∈ 𝐴) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
7	6	3exp 1119	. . . 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 3239	. 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 2111  ∃wrex 3056   ⊆ wss 3897  ∩ ciin 4940  Univcgru 10681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iin 4942  df-br 5090  df-tr 5197  df-iota 6437  df-fv 6489  df-ov 7349  df-gru 10682

This theorem is referenced by: (None)