Theorem mnutrcld 41897

Description: Minimal universes contain the elements of their elements. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
mnutrcld.1	⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
mnutrcld.2	⊢ (𝜑 → 𝑈 ∈ 𝑀)
mnutrcld.3	⊢ (𝜑 → 𝐴 ∈ 𝑈)
mnutrcld.4	⊢ (𝜑 → 𝐵 ∈ 𝐴)

Assertion

Ref	Expression
mnutrcld	⊢ (𝜑 → 𝐵 ∈ 𝑈)

Distinct variable groups:   𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙   𝑈,𝑞,𝑘,𝑚,𝑛,𝑝,𝑙

Allowed substitution hints:   𝜑(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐵(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnutrcld

Step	Hyp	Ref	Expression
1		mnutrcld.1	. 2 ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
2		mnutrcld.2	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3		mnutrcld.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	1, 2, 3	mnuunid 41895	. 2 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
5		mnutrcld.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
6		elssuni 4871	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝐴)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴)
8	1, 2, 4, 7	mnussd 41881	1 ⊢ (𝜑 → 𝐵 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535  df-sn 4562  df-uni 4840

This theorem is referenced by:  mnutrd  41898  mnurndlem2  41900