Theorem mnutrd 45048

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

Hypotheses

Ref	Expression
mnutrd.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))))}
mnutrd.2	⊢ (𝜑 → 𝑈 ∈ 𝑀)

Assertion

Ref	Expression
mnutrd	⊢ (𝜑 → Tr 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)   𝑀(𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑙)

Proof of Theorem mnutrd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnutrd.1	. . . . 5 ⊢ 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		mnutrd.2	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3	2	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑈 ∈ 𝑀)
4		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
5		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑦)
6	1, 3, 4, 5	mnutrcld 45047	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
7	6	ex 418	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈))
8	7	alrimivv 1961	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈))
9		dftr2 5222	. 2 ⊢ (Tr 𝑈 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈))
10	8, 9	sylibr 237	1 ⊢ (𝜑 → Tr 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091   ⊆ wss 3906  𝒫 cpw 4564  ∪ cuni 4874  Tr wtr 5220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-in 3913  df-ss 3923  df-pw 4566  df-sn 4592  df-uni 4875  df-tr 5221

This theorem is used by:  mnugrud  45052