P. 73 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	finexdc 7201*	Decidability of existence, over a finite set and defined by a decidable proposition. (Contributed by Jim Kingdon, 12-Jul-2022.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∃𝑥 ∈ 𝐴 𝜑)
Theorem	dfrex2fin 7202*	Relationship between universal and existential quantifiers over a finite set. Remark in Section 2.2.1 of [Pierik], p. 8. Although Pierik does not mention the decidability condition explicitly, it does say "only finitely many x to check" which means there must be some way of checking each value of x. (Contributed by Jim Kingdon, 11-Jul-2022.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑))
Theorem	elssdc 7203*	Membership in a finite subset of a set with decidable equality is decidable. (Contributed by Jim Kingdon, 11-Feb-2026.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → DECID 𝑋 ∈ 𝐴)
Theorem	eqsndc 7204*	Decidability of equality between a finite subset of a set with decidable equality, and a singleton whose element is an element of the larger set. (Contributed by Jim Kingdon, 15-Feb-2026.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → DECID 𝐴 = {𝑋})
Theorem	infm 7205*	An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.)
⊢ (ω ≼ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
Theorem	infn0 7206	An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
Theorem	inffiexmid 7207*	If any given set is either finite or infinite, excluded middle follows. For another example, 𝒫 1o is not infinite, by pw1ninf 17004, but also cannot be shown to be finite by pw1fin 7211. (Contributed by Jim Kingdon, 15-Jun-2022.)
⊢ (𝑥 ∈ Fin ∨ ω ≼ 𝑥)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	en2eqpr 7208	Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ ((𝐶 ≈ 2o ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ≠ 𝐵 → 𝐶 = {𝐴, 𝐵}))
Theorem	exmidpw 7209	Excluded middle is equivalent to the power set of 1o having two elements. Remark of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.)
⊢ (EXMID ↔ 𝒫 1o ≈ 2o)
Theorem	exmidpweq 7210	Excluded middle is equivalent to the power set of 1o being 2o. (Contributed by Jim Kingdon, 28-Jul-2024.)
⊢ (EXMID ↔ 𝒫 1o = 2o)
Theorem	pw1fin 7211	Excluded middle is equivalent to the power set of 1o being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.)
⊢ (EXMID ↔ 𝒫 1o ∈ Fin)
Theorem	pw1dc0el 7212	Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)
⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
Theorem	exmidpw2en 7213	The power set of a set being equinumerous to set exponentiation with a base of ordinal 2o is equivalent to excluded middle. This is Metamath 100 proof #52. The forward direction uses excluded middle expressed as EXMID to show this equinumerosity. The reverse direction is the one which establishes that power set being equinumerous to set exponentiation implies excluded middle. This resolves the question of whether we will be able to prove this equinumerosity theorem in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.)
⊢ (EXMID ↔ ∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥))
Theorem	ss1o0el1o 7214	Reformulation of ss1o0el1 4332 using 1o instead of {∅}. (Contributed by BJ, 9-Aug-2024.)
⊢ (𝐴 ⊆ 1o → (∅ ∈ 𝐴 ↔ 𝐴 = 1o))
Theorem	pw1dc1 7215	If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.)
⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
Theorem	fientri3 7216	Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴))
Theorem	nnwetri 7217*	A natural number is well-ordered by E. More specifically, this order both satisfies We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
⊢ (𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))
Theorem	onunsnss 7218	Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ⊆ 𝐴)
Theorem	unfiexmid 7219*	If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.)
⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	unsnfi 7220	Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∪ {𝐵}) ∈ Fin)
Theorem	unsnfidcex 7221	The 𝐵 ∈ 𝑉 condition in unsnfi 7220. This is intended to show that unsnfi 7220 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ 𝐴 ∧ (𝐴 ∪ {𝐵}) ∈ Fin) → DECID ¬ 𝐵 ∈ V)
Theorem	unsnfidcel 7222	The ¬ 𝐵 ∈ 𝐴 condition in unsnfi 7220. This is intended to show that unsnfi 7220 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ Fin) → DECID ¬ 𝐵 ∈ 𝐴)
Theorem	unfidisj 7223	The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ Fin)
Theorem	undifdcss 7224*	Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.)
⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵))
Theorem	undifdc 7225*	Union of complementary parts into whole. This is a case where we can strengthen undifss 3608 from subset to equality. (Contributed by Jim Kingdon, 17-Jun-2022.)
⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)))
Theorem	undiffi 7226	Union of complementary parts into whole. This is a case where we can strengthen undifss 3608 from subset to equality. (Contributed by Jim Kingdon, 2-Mar-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)))
Theorem	unfiin 7227	The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
Theorem	prfidisj 7228	A pair is finite if it consists of two unequal sets. For the case where 𝐴 = 𝐵, see snfig 7097. For the cases where one or both is a proper class, see prprc1 3819, prprc2 3820, or prprc 3821. (Contributed by Jim Kingdon, 31-May-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin)
Theorem	prfidceq 7229*	A pair is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 DECID 𝑥 = 𝑦)    ⇒   ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin)
Theorem	tpfidisj 7230	A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin)
Theorem	tpfidceq 7231*	A triple is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 DECID 𝑥 = 𝑦)    ⇒   ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin)
Theorem	fiintim 7232*	If a class is closed under pairwise intersections, then it is closed under nonempty finite intersections. The converse would appear to require an additional condition, such as 𝑥 and 𝑦 not being equal, or 𝐴 having decidable equality. This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐴))
Theorem	xpfi 7233	The Cartesian product of two finite sets is finite. Lemma 8.1.16 of [AczelRathjen], p. 74. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)
Theorem	imaf1fi 7234	The image of a finite set under a one-to-one mapping is finite. (Contributed by Jim Kingdon, 28-Mar-2026.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)
Theorem	3xpfi 7235	The Cartesian product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
⊢ (𝑉 ∈ Fin → ((𝑉 × 𝑉) × 𝑉) ∈ Fin)
Theorem	fisseneq 7236	A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵)
Theorem	phpeqd 7237	Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 7161 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	ssfirab 7238*	A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 DECID 𝜓)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin)
Theorem	ssfidc 7239*	A subset of a finite set is finite if membership in the subset is decidable. (Contributed by Jim Kingdon, 27-May-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐵 ∈ Fin)
Theorem	exmidssfi 7240*	Excluded middle is equivalent to any subset of a finite set being finite. Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 20-Mar-2026.)
⊢ (EXMID ↔ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin))
Theorem	opabfi 7241*	Finiteness of an ordered pair abstraction which is a decidable subset of finite sets. (Contributed by Jim Kingdon, 16-Sep-2025.)
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 DECID 𝜓)    ⇒   ⊢ (𝜑 → 𝑆 ∈ Fin)
Theorem	infidc 7242*	The intersection of two sets is finite if one of them is and the other is decidable. (Contributed by Jim Kingdon, 24-May-2025.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝐴 ∩ 𝐵) ∈ Fin)
Theorem	snon0 7243	An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)
Theorem	fnfi 7244	A version of fnex 5931 for finite sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)
Theorem	fundmfi 7245	The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ Fun 𝐴) → dom 𝐴 ∈ Fin)
Theorem	fundmfibi 7246	A function is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.)
⊢ (Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
Theorem	resfnfinfinss 7247	The restriction of a function to a finite subset of its domain is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) ∈ Fin)
Theorem	residfi 7248	A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
⊢ (( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)
Theorem	relcnvfi 7249	If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin)
Theorem	funrnfi 7250	The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ ((Rel 𝐴 ∧ Fun ◡𝐴 ∧ 𝐴 ∈ Fin) → ran 𝐴 ∈ Fin)
Theorem	f1ofi 7251	If a 1-1 and onto function has a finite domain, its range is finite. (Contributed by Jim Kingdon, 21-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐵 ∈ Fin)
Theorem	f1dmvrnfibi 7252	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7253. (Contributed by AV, 10-Jan-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
Theorem	f1vrnfibi 7253	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 7252. (Contributed by AV, 10-Jan-2020.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
Theorem	iunfidisj 7254*	The finite union of disjoint finite sets is finite. Note that 𝐵 depends on 𝑥, i.e. can be thought of as 𝐵(𝑥). (Contributed by NM, 23-Mar-2006.) (Revised by Jim Kingdon, 7-Oct-2022.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)
Theorem	mapfi 7255	Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ↑𝑚 𝐵) ∈ Fin)
Theorem	elfpw 7256	Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐴 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ Fin))
Theorem	fissfi 7257*	A finite subset of a finite set is a decidable subset. (Contributed by Jim Kingdon, 18-May-2026.)
⊢ ((𝑆 ⊆ 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝑆 ∈ Fin) → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝑆)
Theorem	f1finf1o 7258	Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
Theorem	en1eqsn 7259	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴})
Theorem	en1eqsnbi 7260	A set containing an element has exactly one element iff it is a singleton. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴}))
Theorem	snexxph 7261*	A case where the antecedent of snexg 4319 is not needed. The class {𝑥 ∣ 𝜑} is from dcextest 4726. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
⊢ {{𝑥 ∣ 𝜑}} ∈ V
Theorem	preimaf1ofi 7262	The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.)
⊢ (𝜑 → 𝐶 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ Fin)    ⇒   ⊢ (𝜑 → (◡𝐹 “ 𝐶) ∈ Fin)
Theorem	fidcenumlemim 7263*	Lemma for fidcenum 7267. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.)
⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴))
Theorem	fidcenumlemrks 7264*	Lemma for fidcenum 7267. Induction step for fidcenumlemrk 7265. (Contributed by Jim Kingdon, 20-Oct-2022.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   ⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)    &   ⊢ (𝜑 → 𝐽 ∈ ω)    &   ⊢ (𝜑 → suc 𝐽 ⊆ 𝑁)    &   ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐽)))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
Theorem	fidcenumlemrk 7265*	Lemma for fidcenum 7267. (Contributed by Jim Kingdon, 20-Oct-2022.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   ⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ ω)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ 𝐾) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐾)))
Theorem	fidcenumlemr 7266*	Lemma for fidcenum 7267. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   ⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    ⇒   ⊢ (𝜑 → 𝐴 ∈ Fin)
Theorem	fidcenum 7267*	A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as ∃𝑛 ∈ ω∃𝑓𝑓:𝑛–onto→𝐴 is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.)
⊢ (𝐴 ∈ Fin ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴))
2.6.33  Schroeder-Bernstein Theorem
Theorem	sbthlem1 7268*	Lemma for isbth 7278. (Contributed by NM, 22-Mar-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    ⇒   ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
Theorem	sbthlem2 7269*	Lemma for isbth 7278. (Contributed by NM, 22-Mar-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    ⇒   ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷)
Theorem	sbthlemi3 7270*	Lemma for isbth 7278. (Contributed by NM, 22-Mar-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    ⇒   ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))
Theorem	sbthlemi4 7271*	Lemma for isbth 7278. (Contributed by NM, 27-Mar-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    ⇒   ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
Theorem	sbthlemi5 7272*	Lemma for isbth 7278. (Contributed by NM, 22-Mar-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    &   ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))    ⇒   ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = 𝐴)
Theorem	sbthlemi6 7273*	Lemma for isbth 7278. (Contributed by NM, 27-Mar-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    &   ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))    ⇒   ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵)
Theorem	sbthlem7 7274*	Lemma for isbth 7278. (Contributed by NM, 27-Mar-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    &   ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))    ⇒   ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻)
Theorem	sbthlemi8 7275*	Lemma for isbth 7278. (Contributed by NM, 27-Mar-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    &   ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))    ⇒   ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻)
Theorem	sbthlemi9 7276*	Lemma for isbth 7278. (Contributed by NM, 28-Mar-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    &   ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))    ⇒   ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵)
Theorem	sbthlemi10 7277*	Lemma for isbth 7278. (Contributed by NM, 28-Mar-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    &   ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵)
Theorem	isbth 7278	Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set 𝐴 is smaller (has lower cardinality) than 𝐵 and vice-versa, then 𝐴 and 𝐵 are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 7268 through sbthlemi10 7277; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7277. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. The proof does require the law of the excluded middle which cannot be avoided as shown at exmidsbthr 17042. (Contributed by NM, 8-Jun-1998.)
⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵)
2.6.34  Finitely supported functions
Syntax	cfsupp 7279	Extend class definition to include the predicate to be a finitely supported function.
class finSupp
Definition	df-fsupp 7280*	Define the property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.)
⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
Theorem	relfsupp 7281	The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.)
⊢ Rel finSupp
Theorem	relprcnfsupp 7282	A proper class is never finitely supported. (Contributed by AV, 7-Jun-2019.)
⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 finSupp 𝑍)
Theorem	isfsupp 7283	The property of a class to be a finitely supported function (in relation to a given zero). (Contributed by AV, 23-May-2019.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
Theorem	isfsuppd 7284	Deduction form of isfsupp 7283. (Contributed by SN, 29-Jul-2024.)
⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → Fun 𝑅)    &   ⊢ (𝜑 → (𝑅 supp 𝑍) ∈ Fin)    ⇒   ⊢ (𝜑 → 𝑅 finSupp 𝑍)
Theorem	funisfsupp 7285	The property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.)
⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (𝑅 supp 𝑍) ∈ Fin))
Theorem	fsuppimp 7286	Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019.)
⊢ (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))
Theorem	fsuppimpd 7287	A finitely supported function is a function with a finite support. (Contributed by AV, 6-Jun-2019.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Theorem	fsuppfund 7288	A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	suppeqfsuppbi 7289	If two functions have the same support, one function is finitely supported iff the other one is finitely supported. (Contributed by AV, 30-Jun-2019.)
⊢ (((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍 ↔ 𝐺 finSupp 𝑍)))
Theorem	fsuppxpfi 7290	The cartesian product of two finitely supported functions is finite. (Contributed by AV, 17-Jul-2019.)
⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → ((𝐹 supp 𝑍) × (𝐺 supp 𝑍)) ∈ Fin)
Theorem	fczfsuppd 7291	A constant function with value zero is finitely supported. (Contributed by AV, 30-Jun-2019.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍)
Theorem	0fsupp 7292	The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
⊢ (𝑍 ∈ 𝑉 → ∅ finSupp 𝑍)
Theorem	snopfsuppdc 7293	A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → DECID 𝑌 = 𝑍)    ⇒   ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} finSupp 𝑍)
Theorem	ffsuppbi 7294	Two ways of saying that a function with known codomain is finitely supported. (Contributed by AV, 8-Jul-2019.)
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 finSupp 𝑍 ↔ (◡𝐹 “ (𝑆 ∖ {𝑍})) ∈ Fin)))
Theorem	fsuppcorn 7295	The composition of a 1-1 function with a finitely supported function is finitely supported. The purpose of the (𝐹 supp 𝑍) ⊆ ran 𝐺 condition is to ensure we don't subset the support of the function in such a way as to fun afoul of exmidssfi 7240. (Other alternative conditions might also be sufficient). (Contributed by AV, 28-May-2019.) (Revised by Jim Kingdon, 15-May-2026.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ran 𝐺)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍)
2.6.35  Finite intersections
Syntax	cfi 7296	Extend class notation with the function whose value is the class of finite intersections of the elements of a given set.
class fi
Definition	df-fi 7297*	Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 7300). (Contributed by FL, 27-Apr-2008.)
⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
Theorem	fival 7298*	The set of all the finite intersections of the elements of 𝐴. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})
Theorem	elfi 7299*	Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥))
Theorem	elfi2 7300*	The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))