P. 370 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36901-37000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	meran2 36901	A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜏 ∨ 𝜃) ∨ (𝜒 ∨ (𝜑 ∨ 𝜃))))
Theorem	meran3 36902	A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑))))
Theorem	waj-ax 36903	A single axiom for propositional calculus discovered by Mordchaj Wajsberg (Logical Works, Polish Academy of Sciences, 1977). See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom W on slide 8). (Contributed by Anthony Hart, 13-Aug-2011.)
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ (((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))) ⊼ (𝜑 ⊼ (𝜑 ⊼ 𝜓))))
Theorem	lukshef-ax2 36904	A single axiom for propositional calculus discovered by Jan Lukasiewicz. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom L2 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜑)) ⊼ ((𝜃 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
Theorem	arg-ax 36905	A single axiom for propositional calculus discovered by Ken Harris and Branden Fitelson. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom HF1 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜒 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
21.14.4  Connective Symmetry
Theorem	negsym1 36906	In the paper "On Variable Functors of Propositional Arguments", Lukasiewicz introduced a system that can handle variable connectives. This was done by introducing a variable, marked with a lowercase delta, which takes a wff as input. In the system, "delta 𝜑 " means that "something is true of 𝜑". The expression "delta 𝜑 " can be substituted with ¬ 𝜑, 𝜓 ∧ 𝜑, ∀𝑥𝜑, etc. Later on, Meredith discovered a single axiom, in the form of ( delta delta ⊥ → delta 𝜑 ). This represents the shortest theorem in the extended propositional calculus that cannot be derived as an instance of a theorem in propositional calculus. A symmetry with ¬. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ (¬ ¬ ⊥ → ¬ 𝜑)
Theorem	imsym1 36907	A symmetry with →. See negsym1 36906 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 → (𝜓 → ⊥)) → (𝜓 → 𝜑))
Theorem	bisym1 36908	A symmetry with ↔. See negsym1 36906 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓 ↔ 𝜑))
Theorem	consym1 36909	A symmetry with ∧. See negsym1 36906 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑))
Theorem	dissym1 36910	A symmetry with ∨. See negsym1 36906 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓 ∨ 𝜑))
Theorem	nandsym1 36911	A symmetry with ⊼. See negsym1 36906 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓 ⊼ 𝜑))
Theorem	unisym1 36912	A symmetry with ∀. See negsym1 36906 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ (∀𝑥∀𝑥⊥ → ∀𝑥𝜑)
Theorem	exisym1 36913	A symmetry with ∃. See negsym1 36906 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ (∃𝑥∃𝑥⊥ → ∃𝑥𝜑)
Theorem	unqsym1 36914	A symmetry with ∃!. See negsym1 36906 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)
⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)
Theorem	amosym1 36915	A symmetry with ∃*. See negsym1 36906 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
Theorem	subsym1 36916	A symmetry with [𝑥 / 𝑦]. See negsym1 36906 for more information. (Contributed by Anthony Hart, 11-Sep-2011.)
⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑)
21.15  Mathbox for Chen-Pang He
21.15.1  Ordinal topology
Theorem	ontopbas 36917	An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases)
Theorem	onsstopbas 36918	The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.)
⊢ On ⊆ TopBases
Theorem	onpsstopbas 36919	The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.)
⊢ On ⊊ TopBases
Theorem	ontgval 36920	The topology generated from an ordinal number 𝐵 is suc ∪ 𝐵. (Contributed by Chen-Pang He, 10-Oct-2015.)
⊢ (𝐵 ∈ On → (topGen‘𝐵) = suc ∪ 𝐵)
Theorem	ontgsucval 36921	The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.)
⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴)
Theorem	onsuctop 36922	A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
Theorem	onsuctopon 36923	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 36924	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 36925	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 36926	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc 𝐴 ∈ Conn
Theorem	onsucconn 36927	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn)
Theorem	ordtopconn 36928	An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn))
Theorem	onintopssconn 36929	An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (On ∩ Top) ⊆ Conn
Theorem	onsuct0 36930	A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2)
Theorem	ordtopt0 36931	An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))
Theorem	onsucsuccmpi 36932	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 36933	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 36934	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ Lim 𝐴    ⇒   ⊢ ¬ suc 𝐴 ∈ Comp
Theorem	limsucncmp 36935	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
Theorem	ordcmp 36936	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 36937	The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
⊢ {1o, 2o} ⊆ (On ∩ Haus)
Theorem	onint1 36938	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 36939	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 36940	Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃)))
Theorem	findfvcl 36941*	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 36942*	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 36943*	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 36944	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 36945	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 36946	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 36947	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 36948	Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐵)
Definition	df-gcdOLD 36949*	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 36950	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 36951*	Value of the relation constructed in weiunpo 36954, weiunso 36955, weiunfr 36956, and weiunse 36957. (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 36952*	Lemma for weiunpo 36954, weiunso 36955, weiunfr 36956, and weiunse 36957. (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 36953*	Lemma for weiunfr 36956. (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 36954*	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 36955*	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 36956*	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 36957*	The relation constructed in weiunpo 36954, weiunso 36955, weiunfr 36956, and weiunwe 36958 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 36958*	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 36959*	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 36960*	Axiom of Transitive Containment, derived as a theorem from ax-ext 2735, ax-rep 5239, and ax-inf2 9611. Use ax-tco 36961 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 36961*	The Axiom of Transitive Containment of ZF set theory. It was derived as axtco 36960 above and is therefore redundant if we assume ax-ext 2735, ax-rep 5239 and ax-inf2 9611, 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 9705. 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 36962 and axtco1g 36965, and the weak form ⊢ ∃𝑦(𝑥 ⊆ 𝑦 ∧ Tr 𝑦), see axtco2 36963 and axtco2g 36966. The weak form follows directly from the strong form, see axtco2 36963. But the strong form only follows from the weak form if we allow el 5421 or one of its variants, see axtco1from2 36964. 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 36967 for reasons of syntax. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
Theorem	axtco1 36962*	Strong form of the Axiom of Transitive Containment. See ax-tco 36961 for more information. In particular, this theorem generalizes the statement of ax-tco 36961, 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 36963*	Weak form of the Axiom of Transitive Containment. See ax-tco 36961 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 36964*	Strong form axtco1 36962 of the Axiom of Transitive Containment, derived from the weak form axtco2 36963. See ax-tco 36961 for more information. As written, the proof uses ax-pr 5406 via el 5421, but we could alternatively use ax-pow 5338 via elALT2 5342. Use axtco1 36962 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 36965*	Strong form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 36961 for more information. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))
Theorem	axtco2g 36966*	Weak form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 36961 for more information. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
Theorem	axtcond 36967	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 2404. (Contributed by Matthew House, 6-Apr-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦) → ∀𝑥(𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦))
Theorem	axuntco 36968*	Derivation of ax-un 7734 from ax-tco 36961. Use ax-un 7734 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 36969*	Derivation of ax-nul 5270 from ax-reg 9555 and ax-tco 36961. Use ax-nul 5270 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	elALTtco 36970*	Derivation of el 5421 from ax-tco 36961. Use el 5421 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	tz9.1ctco 36971*	Version of tz9.1c 9700 derived from ax-tco 36961. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
Theorem	tz9.1tco 36972*	Version of tz9.1 9699 derived from ax-tco 36961. (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 36973	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 36974. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ Tr 𝐴) → ∅ ∈ 𝐴)
Theorem	tr0el 36974	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 36975	Extend class notation with the transitive closure of a class. (Contributed by Matthew House, 6-Apr-2026.)
class TC+ 𝐴
Definition	df-ttc 36976*	Transitive closure of a class. Unlike (TC‘𝐴) (see df-tc 9705), 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 37015. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
Theorem	ttceq 36977	Equality theorem for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)
Theorem	ttceqi 36978	Equality inference for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ TC+ 𝐴 = TC+ 𝐵
Theorem	ttceqd 36979	Equality deduction for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → TC+ 𝐴 = TC+ 𝐵)
Theorem	nfttc 36980	Bound-variable hypothesis builder for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥TC+ 𝐴
Theorem	ttcid 36981	The transitive closure contains its argument as a subclass. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 ⊆ TC+ 𝐴
Theorem	ttctr 36982	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ Tr TC+ 𝐴
Theorem	ttctr2 36983	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ TC+ 𝐵 → 𝐴 ⊆ TC+ 𝐵)
Theorem	ttctr3 36984	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴
Theorem	ttcmin 36985	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 36986	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 36987	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 36988	The subclass relationship is inherited by transitive closures. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ⊆ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)
Theorem	ttcel 36989	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 36990	Elements turn into subclasses upon taking transitive closures. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)
Theorem	ttctrid 36991	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 36992	The transitive closure operation is idempotent. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ TC+ 𝐴 = TC+ 𝐴
Theorem	ssttctr 36993	Transitivity of 𝐴 ⊆ TC+ 𝐵 relationship. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ⊆ TC+ 𝐵 ∧ 𝐵 ⊆ TC+ 𝐶) → 𝐴 ⊆ TC+ 𝐶)
Theorem	elttctr 36994	Transitivity of 𝐴 ∈ TC+ 𝐵 relationship. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ TC+ 𝐵 ∧ 𝐵 ∈ TC+ 𝐶) → 𝐴 ∈ TC+ 𝐶)
Theorem	dfttc2g 36995	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 36996	The transitive closure of the empty set is the empty set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ ∅ = ∅
Theorem	ttc00 36997	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 36998	Distribute proper substitution through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ⦋𝐴 / 𝑥⦌TC+ 𝐵 = TC+ ⦋𝐴 / 𝑥⦌𝐵
Theorem	ttcuniun 36999	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 37000*	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)