P. 373 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37201-37300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onsuctop 37201	A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
Theorem	onsuctopon 37202	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 37203	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 37204	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 37205	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc 𝐴 ∈ Conn
Theorem	onsucconn 37206	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn)
Theorem	ordtopconn 37207	An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn))
Theorem	onintopssconn 37208	An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (On ∩ Top) ⊆ Conn
Theorem	onsuct0 37209	A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2)
Theorem	ordtopt0 37210	An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))
Theorem	onsucsuccmpi 37211	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 37212	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 37213	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ Lim 𝐴    ⇒   ⊢ ¬ suc 𝐴 ∈ Comp
Theorem	limsucncmp 37214	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
Theorem	ordcmp 37215	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 37216	The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
⊢ {1o, 2o} ⊆ (On ∩ Haus)
Theorem	onint1 37217	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 37218	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 37219	Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃)))
Theorem	findfvcl 37220*	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 37221*	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 37222*	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 37223	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 37224	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 37225	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 37226	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 37227	Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐵)
Definition	df-gcdOLD 37228*	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 37229	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 37230*	Value of the relation constructed in weiunpo 37233, weiunso 37234, weiunfr 37235, and weiunse 37236. (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 37231*	Lemma for weiunpo 37233, weiunso 37234, weiunfr 37235, and weiunse 37236. (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 37232*	Lemma for weiunfr 37235. (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 37233*	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 37234*	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 37235*	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 37236*	The relation constructed in weiunpo 37233, weiunso 37234, weiunfr 37235, and weiunwe 37237 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 37237*	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 37238*	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 37239*	Axiom of Transitive Containment, derived as a theorem from ax-ext 2733, ax-rep 5232, and ax-inf2 9635. Use ax-tco 37240 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 37240*	The Axiom of Transitive Containment of ZF set theory. It was derived as axtco 37239 above and is therefore redundant if we assume ax-ext 2733, ax-rep 5232 and ax-inf2 9635, 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 9729. 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 37241 and axtco1g 37244, and the weak form ⊢ ∃𝑦(𝑥 ⊆ 𝑦 ∧ Tr 𝑦), see axtco2 37242 and axtco2g 37245. The weak form follows directly from the strong form, see axtco2 37242. But the strong form only follows from the weak form if we allow el 5406 or one of its variants, see axtco1from2 37243. 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 37246 for reasons of syntax. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
Theorem	axtco1 37241*	Strong form of the Axiom of Transitive Containment. See ax-tco 37240 for more information. In particular, this theorem generalizes the statement of ax-tco 37240, 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 37242*	Weak form of the Axiom of Transitive Containment. See ax-tco 37240 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 37243*	Strong form axtco1 37241 of the Axiom of Transitive Containment, derived from the weak form axtco2 37242. See ax-tco 37240 for more information. As written, the proof uses ax-pr 5391 via el 5406, but we could alternatively use ax-pow 5327 via elALT2 5331. Use axtco1 37241 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 37244*	Strong form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 37240 for more information. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))
Theorem	axtco2g 37245*	Weak form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 37240 for more information. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
Theorem	axtcond 37246	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 2402. (Contributed by Matthew House, 6-Apr-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦) → ∀𝑥(𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦))
Theorem	axuntco 37247*	Derivation of ax-un 7749 from ax-tco 37240. Use ax-un 7749 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 37248*	Derivation of ax-nul 5260 from ax-reg 9579 and ax-tco 37240. 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 37249*	Derivation of el 5406 from ax-tco 37240. Use el 5406 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	tz9.1ctco 37250*	Version of tz9.1c 9724 derived from ax-tco 37240. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
Theorem	tz9.1tco 37251*	Version of tz9.1 9723 derived from ax-tco 37240. (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 37252	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 37253. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ Tr 𝐴) → ∅ ∈ 𝐴)
Theorem	tr0el 37253	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 37254	Extend class notation with the transitive closure of a class. (Contributed by Matthew House, 6-Apr-2026.)
class TC+ 𝐴
Definition	df-ttc 37255*	Transitive closure of a class. Unlike (TC‘𝐴) (see df-tc 9729), 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 37294. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
Theorem	ttceq 37256	Equality theorem for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)
Theorem	ttceqi 37257	Equality inference for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ TC+ 𝐴 = TC+ 𝐵
Theorem	ttceqd 37258	Equality deduction for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → TC+ 𝐴 = TC+ 𝐵)
Theorem	nfttc 37259	Bound-variable hypothesis builder for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥TC+ 𝐴
Theorem	ttcid 37260	The transitive closure contains its argument as a subclass. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 ⊆ TC+ 𝐴
Theorem	ttctr 37261	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ Tr TC+ 𝐴
Theorem	ttctr2 37262	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ TC+ 𝐵 → 𝐴 ⊆ TC+ 𝐵)
Theorem	ttctr3 37263	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
Theorem	ttcmin 37264	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 37265	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 37266	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 37267	The subclass relationship is inherited by transitive closures. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ⊆ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)
Theorem	ttcel 37268	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 37269	Elements turn into subclasses upon taking transitive closures. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)
Theorem	ttctrid 37270	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 37271	The transitive closure operation is idempotent. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ TC+ 𝐴 = TC+ 𝐴
Theorem	ssttctr 37272	Transitivity of 𝐴 ⊆ TC+ 𝐵 relationship. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ⊆ TC+ 𝐵 ∧ 𝐵 ⊆ TC+ 𝐶) → 𝐴 ⊆ TC+ 𝐶)
Theorem	elttctr 37273	Transitivity of 𝐴 ∈ TC+ 𝐵 relationship. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ TC+ 𝐵 ∧ 𝐵 ∈ TC+ 𝐶) → 𝐴 ∈ TC+ 𝐶)
Theorem	dfttc2g 37274	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 37275	The transitive closure of the empty set is the empty set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ ∅ = ∅
Theorem	ttc00 37276	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 37277	Distribute proper substitution through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ⦋𝐴 / 𝑥⦌TC+ 𝐵 = TC+ ⦋𝐴 / 𝑥⦌𝐵
Theorem	ttcuniun 37278	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 37279*	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 37280	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 37281	Distribute union of a class through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ ∪ 𝐴 = ∪ TC+ 𝐴
Theorem	ttciun 37282	Distribute indexed union through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 TC+ 𝐵
Theorem	ttcpwss 37283	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 37284	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 37285	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 37286	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 37287	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 37288	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 37289	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 37290	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 37291	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 37302. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 = (TC‘𝐴))
Theorem	ttcwf 37292	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 37293	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 37294	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 37295	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 37303. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ 𝑉 → (𝐴 ≠ ∅ ↔ ∅ ∈ TC+ 𝐴))
Theorem	dfttc4lem1 37296*	Lemma for dfttc4 37298. (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 37297*	Lemma for dfttc4 37298. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}    ⇒   ⊢ (𝐴 ⊆ 𝐵 ∧ Tr 𝐵)
Theorem	dfttc4 37298*	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 37299. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}
Theorem	elttcirr 37299	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 37298 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 37300	The transitive closure of a set is a set, assuming Transitive Containment. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 ∈ V)