P. 369 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36801-36900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onsuctop 36801	A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
Theorem	onsuctopon 36802	One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴))
Theorem	ordtoplem 36803	Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆)    ⇒   ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆))
Theorem	ordtop 36804	An ordinal is a topology iff it is not its supremum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽))
Theorem	onsucconni 36805	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc 𝐴 ∈ Conn
Theorem	onsucconn 36806	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn)
Theorem	ordtopconn 36807	An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn))
Theorem	onintopssconn 36808	An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (On ∩ Top) ⊆ Conn
Theorem	onsuct0 36809	A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2)
Theorem	ordtopt0 36810	An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))
Theorem	onsucsuccmpi 36811	The successor of a successor ordinal number is a compact topology, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 18-Oct-2015.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc suc 𝐴 ∈ Comp
Theorem	onsucsuccmp 36812	The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.)
⊢ (𝐴 ∈ On → suc suc 𝐴 ∈ Comp)
Theorem	limsucncmpi 36813	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ Lim 𝐴    ⇒   ⊢ ¬ suc 𝐴 ∈ Comp
Theorem	limsucncmp 36814	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
Theorem	ordcmp 36815	An ordinal topology is compact iff the underlying set is its supremum (union) only when the ordinal is 1o. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (Ord 𝐴 → (𝐴 ∈ Comp ↔ (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1o)))
Theorem	ssoninhaus 36816	The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
⊢ {1o, 2o} ⊆ (On ∩ Haus)
Theorem	onint1 36817	The ordinal T1 spaces are 1o and 2o, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.)
⊢ (On ∩ Fre) = {1o, 2o}
Theorem	oninhaus 36818	The ordinal Hausdorff spaces are 1o and 2o. (Contributed by Chen-Pang He, 10-Nov-2015.)
⊢ (On ∩ Haus) = {1o, 2o}
21.16  Mathbox for Jeff Hoffman
21.16.1  Inferences for finite induction on generic function values
Theorem	fveleq 36819	Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃)))
Theorem	findfvcl 36820*	Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
⊢ (𝜑 → (𝐹‘∅) ∈ 𝑃)    &   ⊢ (𝑦 ∈ ω → (𝜑 → ((𝐹‘𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃)))    ⇒   ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃))
Theorem	findreccl 36821*	Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)
⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃)    ⇒   ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
Theorem	findabrcl 36822*	Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃)    ⇒   ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃)
21.16.2  gdc.mm
Theorem	nnssi2 36823	Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
⊢ ℕ ⊆ 𝐷    &   ⊢ (𝐵 ∈ ℕ → 𝜑)    &   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓)    ⇒   ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓)
Theorem	nnssi3 36824	Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
⊢ ℕ ⊆ 𝐷    &   ⊢ (𝐶 ∈ ℕ → 𝜑)    &   ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ∧ 𝜑) → 𝜓)    ⇒   ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)
Theorem	nndivsub 36825	Please add description here. (Contributed by Jeff Hoffman, 17-Jun-2008.)
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴 / 𝐶) ∈ ℕ ∧ 𝐴 < 𝐵)) → ((𝐵 / 𝐶) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝐶) ∈ ℕ))
Theorem	nndivlub 36826	A factor of a positive integer cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) ∈ ℕ → 𝐵 ≤ 𝐴))
Syntax	cgcdOLD 36827	Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐵)
Definition	df-gcdOLD 36828*	gcdOLD (𝐴, 𝐵) is the largest positive integer that evenly divides both 𝐴 and 𝐵. (Contributed by Jeff Hoffman, 17-Jun-2008.) (New usage is discouraged.)
⊢ gcdOLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < )
Theorem	ee7.2aOLD 36829	Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as 𝐴 mod 𝐵. Here, just one subtraction step is proved to preserve the gcdOLD. The rec function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 → gcdOLD (𝐴, 𝐵) = gcdOLD (𝐴, (𝐵 − 𝐴))))
21.17  Mathbox for Matthew House
21.17.1  Relations on well-ordered indexed unions
Theorem	weiunval 36830*	Value of the relation constructed in weiunpo 36833, weiunso 36834, weiunfr 36835, and weiunse 36836. (Contributed by Matthew House, 8-Sep-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ (𝐶𝑇𝐷 ↔ ((𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐷 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝐶)𝑅(𝐹‘𝐷) ∨ ((𝐹‘𝐶) = (𝐹‘𝐷) ∧ 𝐶⦋(𝐹‘𝐶) / 𝑥⦌𝑆𝐷))))
Theorem	weiunlem 36831*	Lemma for weiunpo 36833, weiunso 36834, weiunfr 36835, and weiunse 36836. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ (𝜑 → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ∧ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡)))
Theorem	weiunfrlem 36832*	Lemma for weiunfr 36835. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ 𝐸 = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)    &   ⊢ (𝜑 → 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)    &   ⊢ (𝜑 → 𝑟 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐸 ∈ (𝐹 “ 𝑟) ∧ ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅𝐸 ∧ ∀𝑡 ∈ (𝑟 ∩ ⦋𝐸 / 𝑥⦌𝐵)(𝐹‘𝑡) = 𝐸))
Theorem	weiunpo 36833*	A partial ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of partial orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Po 𝐵) → 𝑇 Po ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	weiunso 36834*	A strict ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of strict orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Or 𝐵) → 𝑇 Or ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	weiunfr 36835*	A well-founded relation on an indexed union can be constructed from a well-ordering on its index class and a collection of well-founded relations on its members. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) → 𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	weiunse 36836*	The relation constructed in weiunpo 36833, weiunso 36834, weiunfr 36835, and weiunwe 36837 is set-like if all members of the indexed union are sets. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → 𝑇 Se ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	weiunwe 36837*	A well-ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
⊢ 𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   ⊢ 𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵) → 𝑇 We ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	numiunnum 36838*	An indexed union of sets is numerable if its index set is numerable and there exists a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
⊢ ((𝐴 ∈ dom card ∧ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ dom card)
21.17.2  Axiom of Transitive Containment
Theorem	axtco 36839*	Axiom of Transitive Containment, derived as a theorem from ax-ext 2737, ax-rep 5231, and ax-inf2 9598. Use ax-tco 36840 instead. (Contributed by Matthew House, 6-Apr-2026.) (New usage is discouraged.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
Axiom	ax-tco 36840*	The Axiom of Transitive Containment of ZF set theory. It was derived as axtco 36839 above and is therefore redundant if we assume ax-ext 2737, ax-rep 5231 and ax-inf2 9598, but we state it as a separate axiom here so that its uses can be identified more easily. It states that a transitive set 𝑦 exists that contains a given set 𝑥. In particular, the transitive closure of 𝑥 is a set, since it is a subset of 𝑦, see df-tc 9692. Traditionally, this statement is not counted as an axiom at all, but as a theorem from Replacement and Infinity. In fact, from the transitive closure of 𝑥 we can construct the set of iterated unions of 𝑥 (and vice versa), and Skolem took the existence of the latter set as a motivation for introducing the Axiom of Replacement. But Transitive Containment is strictly weaker than either of those axioms, so many authors identify it as its own axiom when investigating subsystems of ZF, such as Zermelo set theory or finitist set theory. We follow this separation in order to avoid nonessential usage of the stronger axioms. There are two main versions of this axiom that appear in the literature: the strong form ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ Tr 𝑦), see axtco1 36841 and axtco1g 36844, and the weak form ⊢ ∃𝑦(𝑥 ⊆ 𝑦 ∧ Tr 𝑦), see axtco2 36842 and axtco2g 36845. The weak form follows directly from the strong form, see axtco2 36842. But the strong form only follows from the weak form if we allow el 5409 or one of its variants, see axtco1from2 36843. We take the strong form here as the axiom, since it is slightly shorter when expanded to primitive symbols. Yet the weak form turns out to be more suitable for axtcond 36846 for reasons of syntax. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
Theorem	axtco1 36841*	Strong form of the Axiom of Transitive Containment. See ax-tco 36840 for more information. In particular, this theorem generalizes the statement of ax-tco 36840, allowing it to be written with only three variables, since 𝑥 need not be distinct from both 𝑧 and 𝑤. (Contributed by Matthew House, 7-Apr-2026.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
Theorem	axtco2 36842*	Weak form of the Axiom of Transitive Containment. See ax-tco 36840 for more information. In particular, this theorem shows the derivation of the weak form from the strong form. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))
Theorem	axtco1from2 36843*	Strong form axtco1 36841 of the Axiom of Transitive Containment, derived from the weak form axtco2 36842. See ax-tco 36840 for more information. As written, the proof uses ax-pr 5394 via el 5409, but we could alternatively use ax-pow 5326 via elALT2 5330. Use axtco1 36841 instead. (Contributed by Matthew House, 6-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
Theorem	axtco1g 36844*	Strong form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 36840 for more information. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))
Theorem	axtco2g 36845*	Weak form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 36840 for more information. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
Theorem	axtcond 36846	A version of the Axiom of Transitive Containment with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2406. (Contributed by Matthew House, 6-Apr-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦) → ∀𝑥(𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦))
Theorem	axuntco 36847*	Derivation of ax-un 7722 from ax-tco 36840. Use ax-un 7722 instead. (Contributed by Matthew House, 6-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	axnulregtco 36848*	Derivation of ax-nul 5260 from ax-reg 9542 and ax-tco 36840. Use ax-nul 5260 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	elALTtco 36849*	Derivation of el 5409 from ax-tco 36840. Use el 5409 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	tz9.1ctco 36850*	Version of tz9.1c 9687 derived from ax-tco 36840. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
Theorem	tz9.1tco 36851*	Version of tz9.1 9686 derived from ax-tco 36840. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
21.17.3  Transitive closure of a class
Theorem	tr0elw 36852	Every nonempty transitive set contains the empty set ∅ as an element, a consequence of Regularity. If we assume Transitive Containment, then we can omit the 𝐴 ∈ 𝑉 hypothesis, see tr0el 36853. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ Tr 𝐴) → ∅ ∈ 𝐴)
Theorem	tr0el 36853	Every nonempty transitive class contains the empty set ∅ as an element, a consequence of Regularity and Transitive Containment. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ≠ ∅ ∧ Tr 𝐴) → ∅ ∈ 𝐴)
Syntax	cttc 36854	Extend class notation with the transitive closure of a class. (Contributed by Matthew House, 6-Apr-2026.)
class TC+ 𝐴
Definition	df-ttc 36855*	Transitive closure of a class. Unlike (TC‘𝐴) (see df-tc 9692), this definition works even if 𝐴 or its transitive closure is a proper class. Note that unless we assume Transitive Containment, the transitive closure of a set may be a proper class. If we only assume Regularity, then the class of sets whose transitive closure is a set is precisely the class of well-founded sets, see ttcwf3 36894. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
Theorem	ttceq 36856	Equality theorem for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)
Theorem	ttceqi 36857	Equality inference for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ TC+ 𝐴 = TC+ 𝐵
Theorem	ttceqd 36858	Equality deduction for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → TC+ 𝐴 = TC+ 𝐵)
Theorem	nfttc 36859	Bound-variable hypothesis builder for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥TC+ 𝐴
Theorem	ttcid 36860	The transitive closure contains its argument as a subclass. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 ⊆ TC+ 𝐴
Theorem	ttctr 36861	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ Tr TC+ 𝐴
Theorem	ttctr2 36862	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ TC+ 𝐵 → 𝐴 ⊆ TC+ 𝐵)
Theorem	ttctr3 36863	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
Theorem	ttcmin 36864	The transitive closure of 𝐴 is a subclass of every transitive class containing 𝐴. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → TC+ 𝐴 ⊆ 𝐵)
Theorem	ttcexrg 36865	If the transitive closure of a class is a set, then the class is a set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)
Theorem	ttcss 36866	A transitive closure contains the transitive closures of all its subclasses. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ⊆ TC+ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)
Theorem	ttcss2 36867	The subclass relationship is inherited by transitive closures. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ⊆ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)
Theorem	ttcel 36868	A transitive closure contains the transitive closures of all its elements. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ TC+ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)
Theorem	ttcel2 36869	Elements turn into subclasses upon taking transitive closures. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)
Theorem	ttctrid 36870	The transitive closure of a transitive class is the class itself. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (Tr 𝐴 → TC+ 𝐴 = 𝐴)
Theorem	ttcidm 36871	The transitive closure operation is idempotent. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ TC+ 𝐴 = TC+ 𝐴
Theorem	ssttctr 36872	Transitivity of 𝐴 ⊆ TC+ 𝐵 relationship. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ⊆ TC+ 𝐵 ∧ 𝐵 ⊆ TC+ 𝐶) → 𝐴 ⊆ TC+ 𝐶)
Theorem	elttctr 36873	Transitivity of 𝐴 ∈ TC+ 𝐵 relationship. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ TC+ 𝐵 ∧ 𝐵 ∈ TC+ 𝐶) → 𝐴 ∈ TC+ 𝐶)
Theorem	dfttc2g 36874	A shorter expression for the transitive closure of a set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))
Theorem	ttc0 36875	The transitive closure of the empty set is the empty set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ ∅ = ∅
Theorem	ttc00 36876	A class has an empty transitive closure iff it is the empty set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 = ∅ ↔ TC+ 𝐴 = ∅)
Theorem	csbttc 36877	Distribute proper substitution through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ⦋𝐴 / 𝑥⦌TC+ 𝐵 = TC+ ⦋𝐴 / 𝑥⦌𝐵
Theorem	ttcuniun 36878	Relationship between TC+ 𝐴 and TC+ ∪ 𝐴: we can decompose TC+ 𝐴 into the elements of TC+ ∪ 𝐴 plus the elements of 𝐴 itself. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = (TC+ ∪ 𝐴 ∪ 𝐴)
Theorem	ttciunun 36879*	Relationship between TC+ 𝐴 and ∪ 𝑥 ∈ 𝐴TC+ 𝑥: we can decompose TC+ 𝐴 into the elements of ∪ 𝑥 ∈ 𝐴TC+ 𝑥 plus the elements of 𝐴 itself. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)
Theorem	ttcun 36880	Distribute union of two classes through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ (𝐴 ∪ 𝐵) = (TC+ 𝐴 ∪ TC+ 𝐵)
Theorem	ttcuni 36881	Distribute union of a class through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ ∪ 𝐴 = ∪ TC+ 𝐴
Theorem	ttciun 36882	Distribute indexed union through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 TC+ 𝐵
Theorem	ttcpwss 36883	The transitive closure of a power class is contained in the power class of the transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝒫 𝐴 ⊆ 𝒫 TC+ 𝐴
Theorem	ttcsnssg 36884	The transitive closure is contained in the singleton transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 ⊆ TC+ {𝐴})
Theorem	ttcsnidg 36885	The singleton transitive closure contains its argument 𝐴 as an element. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ TC+ {𝐴})
Theorem	ttcsnmin 36886	The singleton transitive closure is the minimal transitive class containing 𝐴 as an element. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ 𝐵 ∧ Tr 𝐵) → TC+ {𝐴} ⊆ 𝐵)
Theorem	ttcsng 36887	Relationship between TC+ {𝐴} and TC+ 𝐴: the former contains the additional element 𝐴. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → TC+ {𝐴} = (TC+ 𝐴 ∪ {𝐴}))
Theorem	ttcsnexg 36888	If the transitive closure of a class is a set, then its singleton transitive closure is a set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ {𝐴} ∈ V)
Theorem	ttcsnexbig 36889	The transitive closure of a set is a set iff its singleton transitive closure is a set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → (TC+ 𝐴 ∈ V ↔ TC+ {𝐴} ∈ V))
Theorem	ttcsntrsucg 36890	The singleton transitive closure of a transitive set is its successor. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ Tr 𝐴) → TC+ {𝐴} = suc 𝐴)
Theorem	dfttc3gw 36891	If the transitive closure of 𝐴 is a set, then its value is (TC‘𝐴). If we assume Transitive Containment, then we can weaken the hypothesis to 𝐴 ∈ 𝑉, see dfttc3g 36902. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 = (TC‘𝐴))
Theorem	ttcwf 36892	A set is well-founded iff its transitive closure is well-founded. As a corollary, the transitive closure of any well-founded set is a set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
Theorem	ttcwf2 36893	If a transitive closure class is a set, then it is well-founded, assuming Regularity. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
Theorem	ttcwf3 36894	The sets whose transitive closures are sets are precisely the well-founded sets, assuming Regularity. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ V ↔ 𝐴 ∈ ∪ (𝑅1 “ On))
Theorem	ttc0elw 36895	If a transitive closure is a set, then it contains ∅ as an element iff it is nonempty, assuming Regularity. If we also assume Transitive Containment, then we can remove the TC+ 𝐴 ∈ 𝑉 hypothesis, see ttc0el 36903. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ 𝑉 → (𝐴 ≠ ∅ ↔ ∅ ∈ TC+ 𝐴))
Theorem	dfttc4lem1 36896*	Lemma for dfttc4 36898. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (((𝐴 ∩ 𝐶) ≠ ∅ ∧ ∀𝑧 ∈ 𝐶 ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷)) → 𝐷 ∈ 𝐵)
Theorem	dfttc4lem2 36897*	Lemma for dfttc4 36898. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}    ⇒   ⊢ (𝐴 ⊆ 𝐵 ∧ Tr 𝐵)
Theorem	dfttc4 36898*	An alternative expression for the transitive closure of a class, assuming Regularity. A set 𝑥 is contained in the transitive closure of 𝐴 iff we can construct an ∈-chain from 𝑥 to an element of 𝐴. This weak definition is primarily useful for proving elttcirr 36899. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}
Theorem	elttcirr 36899	Irreflexivity of 𝐴 ∈ TC+ 𝐵 relationship. This is a consequence of Regularity, but it does not require Transitive Containment. We use the alternative expression dfttc4 36898 to construct a set in which 𝐴 is both ∈-minimal and not ∈-minimal. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ¬ 𝐴 ∈ TC+ 𝐴
Theorem	ttcexg 36900	The transitive closure of a set is a set, assuming Transitive Containment. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 ∈ V)