P. 371 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37001-37100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	filnet 37001*	A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))
21.14  Mathbox for Anthony Hart
21.14.1  Propositional Calculus
Theorem	tb-ax1 37002	The first of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	tb-ax2 37003	The second of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	tb-ax3 37004	The third of three axioms in the Tarski-Bernays axiom system. This axiom, along with ax-mp 5, tb-ax1 37002, and tb-ax2 37003, can be used to derive any theorem or rule that uses only →. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	tbsyl 37005	The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	re1ax2lem 37006	Lemma for re1ax2 37007. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
Theorem	re1ax2 37007	ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 37002 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Theorem	naim1 37008	Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.)
⊢ ((𝜑 → 𝜓) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒)))
Theorem	naim2 37009	Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.)
⊢ ((𝜑 → 𝜓) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑)))
Theorem	naim1i 37010	Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 ⊼ 𝜒)    ⇒   ⊢ (𝜑 ⊼ 𝜒)
Theorem	naim2i 37011	Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 ⊼ 𝜓)    ⇒   ⊢ (𝜒 ⊼ 𝜑)
Theorem	naim12i 37012	Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜃)    &   ⊢ (𝜓 ⊼ 𝜃)    ⇒   ⊢ (𝜑 ⊼ 𝜒)
Theorem	nabi1i 37013	Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ⊼ 𝜒)    ⇒   ⊢ (𝜑 ⊼ 𝜒)
Theorem	nabi2i 37014	Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ⊼ 𝜓)    ⇒   ⊢ (𝜒 ⊼ 𝜑)
Theorem	nabi12i 37015	Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    &   ⊢ (𝜓 ⊼ 𝜃)    ⇒   ⊢ (𝜑 ⊼ 𝜒)
Syntax	w3nand 37016	The double nand.
wff (𝜑 ⊼ 𝜓 ⊼ 𝜒)
Definition	df-3nand 37017	The double nand. This definition allows to express the input of three variables only being false if all three are true. (Contributed by Anthony Hart, 2-Sep-2011.)
⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
Theorem	df3nandALT1 37018	The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)
⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 ⊼ ((𝜓 ⊼ 𝜒) ⊼ (𝜓 ⊼ 𝜒))))
Theorem	df3nandALT2 37019	The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	andnand1 37020	Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ⊼ (𝜑 ⊼ 𝜓 ⊼ 𝜒)))
Theorem	imnand2 37021	An → nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
⊢ ((¬ 𝜑 → 𝜓) ↔ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)))
21.14.2  Predicate Calculus
Theorem	nalfal 37022	Not all sets hold ⊥ as true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ¬ ∀𝑥⊥
Theorem	nexntru 37023	There does not exist a set such that ⊤ is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ¬ ∃𝑥 ¬ ⊤
Theorem	nexfal 37024	There does not exist a set such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ¬ ∃𝑥⊥
Theorem	neufal 37025	There does not exist exactly one set such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ¬ ∃!𝑥⊥
Theorem	neutru 37026	There does not exist exactly one set such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ¬ ∃!𝑥⊤
Theorem	nmotru 37027	There does not exist at most one set such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ¬ ∃*𝑥⊤
Theorem	mofal 37028	There exist at most one set such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ∃*𝑥⊥
Theorem	nrmo 37029	"At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.)
⊢ (𝑥 ∈ 𝐴 → ¬ 𝜑)    ⇒   ⊢ ∃*𝑥 ∈ 𝐴 𝜑
21.14.3  Miscellaneous single axioms
Theorem	meran1 37030	A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜃 ∨ 𝜑) ∨ (𝜒 ∨ (𝜏 ∨ 𝜑))))
Theorem	meran2 37031	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 37032	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 37033	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 37034	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 37035	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 37036	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 37037	A symmetry with →. See negsym1 37036 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 → (𝜓 → ⊥)) → (𝜓 → 𝜑))
Theorem	bisym1 37038	A symmetry with ↔. See negsym1 37036 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓 ↔ 𝜑))
Theorem	consym1 37039	A symmetry with ∧. See negsym1 37036 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑))
Theorem	dissym1 37040	A symmetry with ∨. See negsym1 37036 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓 ∨ 𝜑))
Theorem	nandsym1 37041	A symmetry with ⊼. See negsym1 37036 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓 ⊼ 𝜑))
Theorem	unisym1 37042	A symmetry with ∀. See negsym1 37036 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ (∀𝑥∀𝑥⊥ → ∀𝑥𝜑)
Theorem	exisym1 37043	A symmetry with ∃. See negsym1 37036 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)
⊢ (∃𝑥∃𝑥⊥ → ∃𝑥𝜑)
Theorem	unqsym1 37044	A symmetry with ∃!. See negsym1 37036 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)
⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)
Theorem	amosym1 37045	A symmetry with ∃*. See negsym1 37036 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
Theorem	subsym1 37046	A symmetry with [𝑥 / 𝑦]. See negsym1 37036 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 37047	An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases)
Theorem	onsstopbas 37048	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 37049	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 37050	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 37051	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 37052	A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top)
Theorem	onsuctopon 37053	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 37054	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 37055	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 37056	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc 𝐴 ∈ Conn
Theorem	onsucconn 37057	A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn)
Theorem	ordtopconn 37058	An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn))
Theorem	onintopssconn 37059	An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
⊢ (On ∩ Top) ⊆ Conn
Theorem	onsuct0 37060	A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2)
Theorem	ordtopt0 37061	An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))
Theorem	onsucsuccmpi 37062	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 37063	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 37064	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ Lim 𝐴    ⇒   ⊢ ¬ suc 𝐴 ∈ Comp
Theorem	limsucncmp 37065	The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
Theorem	ordcmp 37066	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 37067	The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
⊢ {1o, 2o} ⊆ (On ∩ Haus)
Theorem	onint1 37068	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 37069	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 37070	Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃)))
Theorem	findfvcl 37071*	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 37072*	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 37073*	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 37074	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 37075	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 37076	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 37077	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 37078	Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐵)
Definition	df-gcdOLD 37079*	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 37080	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 37081*	Value of the relation constructed in weiunpo 37084, weiunso 37085, weiunfr 37086, and weiunse 37087. (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 37082*	Lemma for weiunpo 37084, weiunso 37085, weiunfr 37086, and weiunse 37087. (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 37083*	Lemma for weiunfr 37086. (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 37084*	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 37085*	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 37086*	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 37087*	The relation constructed in weiunpo 37084, weiunso 37085, weiunfr 37086, and weiunwe 37088 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 37088*	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 37089*	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 37090*	Axiom of Transitive Containment, derived as a theorem from ax-ext 2732, ax-rep 5232, and ax-inf2 9620. Use ax-tco 37091 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 37091*	The Axiom of Transitive Containment of ZF set theory. It was derived as axtco 37090 above and is therefore redundant if we assume ax-ext 2732, ax-rep 5232 and ax-inf2 9620, 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 9714. 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 37092 and axtco1g 37095, and the weak form ⊢ ∃𝑦(𝑥 ⊆ 𝑦 ∧ Tr 𝑦), see axtco2 37093 and axtco2g 37096. The weak form follows directly from the strong form, see axtco2 37093. But the strong form only follows from the weak form if we allow el 5413 or one of its variants, see axtco1from2 37094. 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 37097 for reasons of syntax. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
Theorem	axtco1 37092*	Strong form of the Axiom of Transitive Containment. See ax-tco 37091 for more information. In particular, this theorem generalizes the statement of ax-tco 37091, 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 37093*	Weak form of the Axiom of Transitive Containment. See ax-tco 37091 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 37094*	Strong form axtco1 37092 of the Axiom of Transitive Containment, derived from the weak form axtco2 37093. See ax-tco 37091 for more information. As written, the proof uses ax-pr 5398 via el 5413, but we could alternatively use ax-pow 5330 via elALT2 5334. Use axtco1 37092 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 37095*	Strong form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 37091 for more information. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))
Theorem	axtco2g 37096*	Weak form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 37091 for more information. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
Theorem	axtcond 37097	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 2401. (Contributed by Matthew House, 6-Apr-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦) → ∀𝑥(𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦))
Theorem	axuntco 37098*	Derivation of ax-un 7736 from ax-tco 37091. Use ax-un 7736 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 37099*	Derivation of ax-nul 5263 from ax-reg 9564 and ax-tco 37091. Use ax-nul 5263 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	elALTtco 37100*	Derivation of el 5413 from ax-tco 37091. Use el 5413 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦