Theorem mnussd 44500

Description: Minimal universes are closed under subsets. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
mnussd.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))))}
mnussd.2	⊢ (𝜑 → 𝑈 ∈ 𝑀)
mnussd.3	⊢ (𝜑 → 𝐴 ∈ 𝑈)
mnussd.4	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
mnussd	⊢ (𝜑 → 𝐵 ∈ 𝑈)

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

Dummy variables 𝑤 𝑓 𝑖 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnussd.1	. . . 4 ⊢ 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		mnussd.2	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3		mnussd.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	1, 2, 3	mnuop123d 44499	. . 3 ⊢ (𝜑 → (𝒫 𝐴 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
5	4	simpld 494	. 2 ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝑈)
6		mnussd.4	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
7	3, 6	sselpwd 5273	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴)
8	5, 7	sseldd 3934	1 ⊢ (𝜑 → 𝐵 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃wrex 3060   ⊆ wss 3901  𝒫 cpw 4554  ∪ cuni 4863

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 2115  ax-9 2123  ax-ext 2708  ax-sep 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-in 3908  df-ss 3918  df-pw 4556  df-uni 4864

This theorem is referenced by:  mnuss2d  44501  mnu0eld  44502  mnusnd  44505  mnuprssd  44506  mnuprdlem4  44512  mnutrcld  44516