Theorem gruiin 10739

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 1914	. . 3 ⊢ Ⅎ𝑥 𝑈 ∈ Univ
2		nfii1 4989	. . . 4 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵
3	2	nfel1 2908	. . 3 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈
4		iinss2 5016	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)
5		gruss 10725	. . . . . 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 3242	. 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 2109  ∃wrex 3053   ⊆ wss 3911  ∩ ciin 4952  Univcgru 10719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iin 4954  df-br 5103  df-tr 5210  df-iota 6452  df-fv 6507  df-ov 7372  df-gru 10720

This theorem is referenced by: (None)