P. 449 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44801-44900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mnringvald 44801*	Value of the monoid ring function. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑉 = (𝑅 freeLMod 𝐴)    &   ⊢ 𝐵 = (Base‘𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))
Theorem	mnringnmulrd 44802	Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (Revised by AV, 1-Nov-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (.r‘ndx)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ 𝑉 = (𝑅 freeLMod 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))
Theorem	mnringbased 44803	The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ 𝑉 = (𝑅 freeLMod 𝐴)    &   ⊢ 𝐵 = (Base‘𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝐹))
Theorem	mnringbaserd 44804	The base set of a monoid ring. Converse of mnringbased 44803. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ 𝑉 = (𝑅 freeLMod 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝑉))
Theorem	mnringelbased 44805	Membership in the base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ 𝐶 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝐶 ↑m 𝐴) ∧ 𝑋 finSupp 0 )))
Theorem	mnringbasefd 44806	Elements of a monoid ring are functions. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ 𝐶 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋:𝐴⟶𝐶)
Theorem	mnringbasefsuppd 44807	Elements of a monoid ring are finitely supported. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 finSupp 0 )
Theorem	mnringaddgd 44808	The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ 𝑉 = (𝑅 freeLMod 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (+g‘𝑉) = (+g‘𝐹))
Theorem	mnring0gd 44809	The additive identity of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ 𝑉 = (𝑅 freeLMod 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (0g‘𝑉) = (0g‘𝐹))
Theorem	mnring0g2d 44810	The additive identity of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴 × { 0 }) = (0g‘𝐹))
Theorem	mnringmulrd 44811*	The ring product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 ))))) = (.r‘𝐹))
Theorem	mnringscad 44812	The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
Theorem	mnringvscad 44813	The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑉 = (𝑅 freeLMod 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ( ·𝑠 ‘𝑉) = ( ·𝑠 ‘𝐹))
Theorem	mnringlmodd 44814	Monoid rings are left modules. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝐹 ∈ LMod)
Theorem	mnringmulrvald 44815*	Value of multiplication in a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ ∙ = (.r‘𝑅)    &   ⊢ 𝟎 = (0g‘𝑅)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ · = (.r‘𝐹)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑋‘𝑎) ∙ (𝑌‘𝑏)), 𝟎 )))))
Theorem	mnringmulrcld 44816	Monoid rings are closed under multiplication. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐴 = (Base‘𝑀)    &   ⊢ · = (.r‘𝐹)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
21.38.3  Shorter primitive equivalent of ax-groth
21.38.3.1  Grothendieck universes are closed under collection
Theorem	gru0eld 44817	A nonempty Grothendieck universe contains the empty set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Univ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐺)    ⇒   ⊢ (𝜑 → ∅ ∈ 𝐺)
Theorem	grusucd 44818	Grothendieck universes are closed under ordinal successor. (Contributed by Rohan Ridenour, 9-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Univ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐺)    ⇒   ⊢ (𝜑 → suc 𝐴 ∈ 𝐺)
Theorem	r1rankcld 44819	Any rank of the cumulative hierarchy is closed under the rank function. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ (𝑅1‘𝑅))    ⇒   ⊢ (𝜑 → (rank‘𝐴) ∈ (𝑅1‘𝑅))
Theorem	grur1cld 44820	Grothendieck universes are closed under the cumulative hierarchy function. (Contributed by Rohan Ridenour, 8-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Univ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐺)    ⇒   ⊢ (𝜑 → (𝑅1‘𝐴) ∈ 𝐺)
Theorem	grurankcld 44821	Grothendieck universes are closed under the rank function. (Contributed by Rohan Ridenour, 9-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Univ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐺)    ⇒   ⊢ (𝜑 → (rank‘𝐴) ∈ 𝐺)
Theorem	grurankrcld 44822	If a Grothendieck universe contains a set's rank, it contains that set. (Contributed by Rohan Ridenour, 9-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Univ)    &   ⊢ (𝜑 → (rank‘𝐴) ∈ 𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐺)
Theorem	gruscottcld 44823	If a Grothendieck universe contains an element of a Scott's trick set, it contains the Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Univ)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐺)    &   ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴)    ⇒   ⊢ (𝜑 → Scott 𝐴 ∈ 𝐺)
Syntax	ccoll 44824	Extend class notation with the collection operation.
class (𝐹 Coll 𝐴)
Definition	df-coll 44825*	Define the collection operation. This is similar to the image set operation “, but it uses Scott's trick to ensure the output is always a set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
Theorem	dfcoll2 44826*	Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}
Theorem	colleq12d 44827	Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))
Theorem	colleq1 44828	Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝐹 = 𝐺 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐴))
Theorem	colleq2 44829	Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝐴 = 𝐵 → (𝐹 Coll 𝐴) = (𝐹 Coll 𝐵))
Theorem	nfcoll 44830	Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹 Coll 𝐴)
Theorem	collexd 44831	The output of the collection operation is a set if the second input is. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 Coll 𝐴) ∈ V)
Theorem	cpcolld 44832*	Property of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑥𝐹𝑦)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
Theorem	cpcoll2d 44833*	cpcolld 44832 with an extra existential quantifier. (Contributed by Rohan Ridenour, 12-Aug-2023.)
⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑦 𝑥𝐹𝑦)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
Theorem	grucollcld 44834	A Grothendieck universe contains the output of a collection operation whenever its left input is a relation on the universe, and its right input is in the universe. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Univ)    &   ⊢ (𝜑 → 𝐹 ⊆ (𝐺 × 𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐺)    ⇒   ⊢ (𝜑 → (𝐹 Coll 𝐴) ∈ 𝐺)
21.38.3.2  Minimal universes
Theorem	ismnu 44835*	The hypothesis of this theorem defines a class M of sets that we temporarily call "minimal universes", and which will turn out in grumnueq 44861 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 44873, except for the 𝑥 ∈ 𝑦 clause. A minimal universe is closed under subsets (mnussd 44837), powersets (mnupwd 44841), and an operation which is similar to a combination of collection and union (mnuop3d 44845), from which closure under pairing (mnuprd 44850), unions (mnuunid 44851), and function ranges (mnurnd 44857) can be deduced, from which equivalence with Grothendieck universes (grumnueq 44861) can be deduced. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    ⇒   ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ 𝑀 ↔ ∀𝑧 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))))
Theorem	mnuop123d 44836*	Operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝒫 𝐴 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
Theorem	mnussd 44837*	Minimal universes are closed under subsets. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑈)
Theorem	mnuss2d 44838*	mnussd 44837 with arguments provided with an existential quantifier. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 𝐴 ⊆ 𝑥)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝑈)
Theorem	mnu0eld 44839*	A nonempty minimal universe contains the empty set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ∅ ∈ 𝑈)
Theorem	mnuop23d 44840*	Second and third operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹) → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
Theorem	mnupwd 44841*	Minimal universes are closed under powersets. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈)
Theorem	mnusnd 44842*	Minimal universes are closed under singletons. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    ⇒   ⊢ (𝜑 → {𝐴} ∈ 𝑈)
Theorem	mnuprssd 44843*	A minimal universe contains pairs of subsets of an element of the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Theorem	mnuprss2d 44844*	Special case of mnuprssd 44843. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ 𝐴 ⊆ 𝐶    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Theorem	mnuop3d 44845*	Third operation of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
Theorem	mnuprdlem1 44846*	Lemma for mnuprd 44850. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝑤)
Theorem	mnuprdlem2 44847*	Lemma for mnuprd 44850. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → ¬ 𝐴 = ∅)    &   ⊢ (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑤)
Theorem	mnuprdlem3 44848*	Lemma for mnuprd 44850. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   ⊢ Ⅎ𝑖𝜑    ⇒   ⊢ (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑣 ∈ 𝐹 𝑖 ∈ 𝑣)
Theorem	mnuprdlem4 44849*	Lemma for mnuprd 44850. General case. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → ¬ 𝐴 = ∅)    ⇒   ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Theorem	mnuprd 44850*	Minimal universes are closed under pairing. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    ⇒   ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Theorem	mnuunid 44851*	Minimal universes are closed under union. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
Theorem	mnuund 44852*	Minimal universes are closed under binary unions. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈)
Theorem	mnutrcld 44853*	Minimal universes contain the elements of their elements. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑈)
Theorem	mnutrd 44854*	Minimal universes are transitive. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    ⇒   ⊢ (𝜑 → Tr 𝑈)
Theorem	mnurndlem1 44855*	Lemma for mnurnd 44857. (Contributed by Rohan Ridenour, 12-Aug-2023.)
⊢ (𝜑 → 𝐹:𝐴⟶𝑈)    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})𝑖 ∈ 𝑣 → ∃𝑢 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})(𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))    ⇒   ⊢ (𝜑 → ran 𝐹 ⊆ 𝑤)
Theorem	mnurndlem2 44856*	Lemma for mnurnd 44857. Deduction theorem input. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑈)    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝜑 → ran 𝐹 ∈ 𝑈)
Theorem	mnurnd 44857*	Minimal universes contain ranges of functions from an element of the universe to the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑈)    ⇒   ⊢ (𝜑 → ran 𝐹 ∈ 𝑈)
Theorem	mnugrud 44858*	Minimal universes are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝑈 ∈ 𝑀)    ⇒   ⊢ (𝜑 → 𝑈 ∈ Univ)
Theorem	grumnudlem 44859*	Lemma for grumnud 44860. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝐺 ∈ Univ)    &   ⊢ 𝐹 = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺))    &   ⊢ ((𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺) → (𝑖𝐹ℎ ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))    &   ⊢ ((ℎ ∈ (𝐹 Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝑀)
Theorem	grumnud 44860*	Grothendieck universes are minimal universes. (Contributed by Rohan Ridenour, 12-Aug-2023.)
⊢ 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))))}    &   ⊢ (𝜑 → 𝐺 ∈ Univ)    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝑀)
Theorem	grumnueq 44861*	The class of Grothendieck universes is equal to the class of minimal universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ Univ = {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))))}
21.38.3.3  Primitive equivalent of ax-groth
Theorem	expandan 44862	Expand conjunction to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) ↔ ¬ (𝜓 → ¬ 𝜃))
Theorem	expandexn 44863	Expand an existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥𝜓)
Theorem	expandral 44864	Expand a restricted universal quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
Theorem	expandrexn 44865	Expand a restricted existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
Theorem	expandrex 44866	Expand a restricted existential quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜓))
Theorem	expanduniss 44867*	Expand ∪ 𝐴 ⊆ 𝐵 to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
Theorem	ismnuprim 44868*	Express the predicate on 𝑈 in ismnu 44835 using only primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ (∀𝑧 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ ∀𝑧(𝑧 ∈ 𝑈 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑈 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))))))
Theorem	rr-grothprimbi 44869*	Express "every set is contained in a Grothendieck universe" using only primitives. The right side (without the outermost universal quantifier) is proven as rr-grothprim 44874. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ (∀𝑥∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∀𝑥 ¬ ∀𝑦(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))))
Theorem	inagrud 44870	Inaccessible levels of the cumulative hierarchy are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ (𝜑 → 𝐼 ∈ Inacc)    ⇒   ⊢ (𝜑 → (𝑅1‘𝐼) ∈ Univ)
Theorem	inaex 44871*	Assuming the Tarski-Grothendieck axiom, every ordinal is contained in an inaccessible ordinal. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ Inacc 𝐴 ∈ 𝑥)
Theorem	gruex 44872*	Assuming the Tarski-Grothendieck axiom, every set is contained in a Grothendieck universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ ∃𝑦 ∈ Univ 𝑥 ∈ 𝑦
Theorem	rr-groth 44873*	An equivalent of ax-groth 10796 using only simple defined symbols. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑓∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑦 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
Theorem	rr-grothprim 44874*	An equivalent of ax-groth 10796 using only primitives. This uses only 123 symbols, which is significantly less than the previous record of 163 established by grothprim 10807 (which uses some defined symbols, and requires 229 symbols if expanded to primitives). (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ ¬ ∀𝑦(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))))))
Theorem	ismnushort 44875*	Express the predicate on 𝑈 and 𝑧 in ismnu 44835 in a shorter form while avoiding complicated definitions. (Contributed by Rohan Ridenour, 10-Oct-2024.)
⊢ (∀𝑓 ∈ 𝒫 𝑈∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ (𝑈 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤)) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
Theorem	dfuniv2 44876*	Alternative definition of Univ using only simple defined symbols. (Contributed by Rohan Ridenour, 10-Oct-2024.)
⊢ Univ = {𝑦 ∣ ∀𝑧 ∈ 𝑦 ∀𝑓 ∈ 𝒫 𝑦∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ (𝑦 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤))}
Theorem	rr-grothshortbi 44877*	Express "every set is contained in a Grothendieck universe" in a short form while avoiding complicated definitions. (Contributed by Rohan Ridenour, 8-Oct-2024.)
⊢ (∀𝑥∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∀𝑥∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∀𝑓 ∈ 𝒫 𝑦∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ (𝑦 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤))))
Theorem	rr-grothshort 44878*	A shorter equivalent of ax-groth 10796 than rr-groth 44873 using a few more simple defined symbols. (Contributed by Rohan Ridenour, 8-Oct-2024.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∀𝑓 ∈ 𝒫 𝑦∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ (𝑦 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤)))
21.39  Mathbox for Steve Rodriguez
21.39.1  Miscellanea
Theorem	nanorxor 44879	'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.)
⊢ ((𝜑 ⊼ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ⊻ 𝜓)))
Theorem	undisjrab 44880	Union of two disjoint restricted class abstractions; compare unrab 4270. (Contributed by Steve Rodriguez, 28-Feb-2020.)
⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = ∅ ↔ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ⊻ 𝜓)})
Theorem	iso0 44881	The empty set is an 𝑅, 𝑆 isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
⊢ ∅ Isom 𝑅, 𝑆 (∅, ∅)
Theorem	ssrecnpr 44882	ℝ is a subset of both ℝ and ℂ. (Contributed by Steve Rodriguez, 22-Nov-2015.)
⊢ (𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆)
Theorem	seff 44883	Let set 𝑆 be the real or complex numbers. Then the exponential function restricted to 𝑆 is a mapping from 𝑆 to 𝑆. (Contributed by Steve Rodriguez, 6-Nov-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    ⇒   ⊢ (𝜑 → (exp ↾ 𝑆):𝑆⟶𝑆)
Theorem	sblpnf 44884	The infinity ball in the absolute value metric is just the whole space. 𝑆 analogue of blpnf 24515. (Contributed by Steve Rodriguez, 8-Nov-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆))    ⇒   ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆)
Theorem	prmunb2 44885*	The primes are unbounded. This generalizes prmunb 16964 to real 𝐴 with arch 12492 and lttrd 11359: every real is less than some positive integer, itself less than some prime. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝐴 ∈ ℝ → ∃𝑝 ∈ ℙ 𝐴 < 𝑝)
21.39.2  Ratio test for infinite series convergence and divergence
Theorem	dvgrat 44886*	Ratio test for divergence of a complex infinite series. See e.g. remark "if (abs‘((𝑎‘(𝑛 + 1)) / (𝑎‘𝑛))) ≥ 1 for all large n..." in https://en.wikipedia.org/wiki/Ratio_test#The_test. (Contributed by Steve Rodriguez, 28-Feb-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ≠ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ≤ (abs‘(𝐹‘(𝑘 + 1))))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∉ dom ⇝ )
Theorem	cvgdvgrat 44887*	Ratio test for convergence and divergence of a complex infinite series. If the ratio 𝑅 of the absolute values of successive terms in an infinite sequence 𝐹 converges to less than one, then the infinite sum of the terms of 𝐹 converges to a complex number; and if 𝑅 converges greater then the sum diverges. This combined form of cvgrat 15927 and dvgrat 44886 directly uses the limit of the ratio. (It also demonstrates how to use climi2 15552 and absltd 15473 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191 15473, and how to use r19.29a 3173 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 3166 at https://groups.google.com/g/metamath/c/2RPikOiXLMo 3166.) (Contributed by Steve Rodriguez, 28-Feb-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ≠ 0)    &   ⊢ 𝑅 = (𝑘 ∈ 𝑊 ↦ (abs‘((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘))))    &   ⊢ (𝜑 → 𝑅 ⇝ 𝐿)    &   ⊢ (𝜑 → 𝐿 ≠ 1)    ⇒   ⊢ (𝜑 → (𝐿 < 1 ↔ seq𝑀( + , 𝐹) ∈ dom ⇝ ))
Theorem	radcnvrat 44888*	Let 𝐿 be the limit, if one exists, of the ratio (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))) (as in the ratio test cvgdvgrat 44887) as 𝑘 increases. Then the radius of convergence of power series Σ𝑛 ∈ ℕ0((𝐴‘𝑛) · (𝑥↑𝑛)) is (1 / 𝐿) if 𝐿 is nonzero. Proof "The limit involved in the ratio test..." in https://en.wikipedia.org/wiki/Radius_of_convergence 44887 —a few lines that evidently hide quite an involved process to confirm. (Contributed by Steve Rodriguez, 8-Mar-2020.)
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐷 = (𝑘 ∈ ℕ0 ↦ (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))))    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴‘𝑘) ≠ 0)    &   ⊢ (𝜑 → 𝐷 ⇝ 𝐿)    &   ⊢ (𝜑 → 𝐿 ≠ 0)    ⇒   ⊢ (𝜑 → 𝑅 = (1 / 𝐿))
21.39.3  Multiples
Theorem	reldvds 44889	The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ Rel ∥
Theorem	nznngen 44890	All positive integers in the set of multiples of n, nℤ, are the absolute value of n or greater. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (( ∥ “ {𝑁}) ∩ ℕ) ⊆ (ℤ≥‘(abs‘𝑁)))
Theorem	nzss 44891	The set of multiples of m, mℤ, is a subset of those of n, nℤ, iff n divides m. Lemma 2.1(a) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑀))
Theorem	nzin 44892	The intersection of the set of multiples of m, mℤ, and those of n, nℤ, is the set of multiples of their least common multiple. Roughly Lemma 2.1(c) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5 and Problem 1(b) of https://people.math.binghamton.edu/mazur/teach/40107/40107h16sol.pdf p. 1, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))
Theorem	nzprmdif 44893	Subtract one prime's multiples from an unequal prime's. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝜑 → 𝑀 ∈ ℙ)    &   ⊢ (𝜑 → 𝑁 ∈ ℙ)    &   ⊢ (𝜑 → 𝑀 ≠ 𝑁)    ⇒   ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)})))
Theorem	hashnzfz 44894	Special case of hashdvds 16824: the count of multiples in nℤ restricted to an interval. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘(𝐽 − 1)))    ⇒   ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (𝐽...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁))))
Theorem	hashnzfz2 44895	Special case of hashnzfz 44894: the count of multiples in nℤ, n greater than one, restricted to an interval starting at two. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    ⇒   ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = (⌊‘(𝐾 / 𝑁)))
Theorem	hashnzfzclim 44896*	As the upper bound 𝐾 of the constraint interval (𝐽...𝐾) in hashnzfz 44894 increases, the resulting count of multiples tends to (𝐾 / 𝑀) —that is, there are approximately (𝐾 / 𝑀) multiples of 𝑀 in a finite interval of integers. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑘 ∈ (ℤ≥‘(𝐽 − 1)) ↦ ((♯‘(( ∥ “ {𝑀}) ∩ (𝐽...𝑘))) / 𝑘)) ⇝ (1 / 𝑀))
21.39.4  Function operations
Theorem	caofcan 44897*	Transfer a cancellation law like mulcan 11839 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑇)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))    ⇒   ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f 𝑅𝐻) ↔ 𝐺 = 𝐻))
Theorem	ofsubid 44898	Function analogue of subid 11465. (Contributed by Steve Rodriguez, 5-Nov-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∘f − 𝐹) = (𝐴 × {0}))
Theorem	ofmul12 44899	Function analogue of mul12 11363. (Contributed by Steve Rodriguez, 13-Nov-2015.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐺:𝐴⟶ℂ ∧ 𝐻:𝐴⟶ℂ)) → (𝐹 ∘f · (𝐺 ∘f · 𝐻)) = (𝐺 ∘f · (𝐹 ∘f · 𝐻)))
Theorem	ofdivrec 44900	Function analogue of divrec 11876, a division analogue of ofnegsub 12207. (Contributed by Steve Rodriguez, 3-Nov-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶(ℂ ∖ {0})) → (𝐹 ∘f · ((𝐴 × {1}) ∘f / 𝐺)) = (𝐹 ∘f / 𝐺))