Theorem mnutrd 44015

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 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑈 ∈ 𝑀)
4		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
5		simprl 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑦)
6	1, 3, 4, 5	mnutrcld 44014	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
7	6	ex 411	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈))
8	7	alrimivv 1924	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈))
9		dftr2 5273	. 2 ⊢ (Tr 𝑈 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈))
10	8, 9	sylibr 233	1 ⊢ (𝜑 → Tr 𝑈)

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1532   = wceq 1534   ∈ wcel 2100  {cab 2706  ∀wral 3054  ∃wrex 3063   ⊆ wss 3957  𝒫 cpw 4608  ∪ cuni 4916  Tr wtr 5271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700  ax-sep 5305

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1537  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-in 3964  df-ss 3974  df-pw 4610  df-sn 4635  df-uni 4917  df-tr 5272

This theorem is referenced by:  mnugrud  44019