P. 359 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	faclim 35801*	An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))    ⇒   ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))
Theorem	iprodfac 35802*	An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.)
⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘))))
Theorem	faclim2 35803*	Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀))))    ⇒   ⊢ (𝑀 ∈ ℕ0 → 𝐹 ⇝ 1)
21.11.8  Greatest common divisor and divisibility
Theorem	gcd32 35804	Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = ((𝐴 gcd 𝐶) gcd 𝐵))
Theorem	gcdabsorb 35805	Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐵) = (𝐴 gcd 𝐵))
21.11.9  Properties of relationships
Theorem	dftr6 35806	A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
Theorem	coep 35807*	Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥)
Theorem	coepr 35808*	Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵)
Theorem	dffr5 35809	A quantifier-free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)))
Theorem	dfso2 35810	Quantifier-free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))))
Theorem	br8 35811*	Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎))    &   ⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)}    ⇒   ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜌))
Theorem	br6 35812*	Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ 𝜑)}    ⇒   ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄) ∧ (𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄)) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩𝑅⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ 𝜁))
Theorem	br4 35813*	Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)}    ⇒   ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜏))
Theorem	cnvco1 35814	Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴)
Theorem	cnvco2 35815	Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴)
Theorem	eldm3 35816	Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)
⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)
Theorem	elrn3 35817	Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)
Theorem	pocnv 35818	The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴)
Theorem	socnv 35819	The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴)
Theorem	elintfv 35820*	Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ 𝑋 ∈ V    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))
21.11.10  Properties of functions and mappings
Theorem	funpsstri 35821	A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹))
Theorem	fundmpss 35822	If a class 𝐹 is a proper subset of a function 𝐺, then dom 𝐹 ⊊ dom 𝐺. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ (Fun 𝐺 → (𝐹 ⊊ 𝐺 → dom 𝐹 ⊊ dom 𝐺))
Theorem	funsseq 35823	Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺 ↔ 𝐹 ⊆ 𝐺))
Theorem	fununiq 35824	The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶))
Theorem	funbreq 35825	An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶))
Theorem	br1steq 35826	Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴)
Theorem	br2ndeq 35827	Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵)
Theorem	dfdm5 35828	Definition of domain in terms of 1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
Theorem	dfrn5 35829	Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
Theorem	opelco3 35830	Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.)
⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
Theorem	elima4 35831	Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.)
⊢ (𝐴 ∈ (𝑅 “ 𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅)
Theorem	fv1stcnv 35832	The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)
Theorem	fv2ndcnv 35833	The value of the converse of 2nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)
21.11.11  Ordinal numbers
Theorem	elpotr 35834*	A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.)
⊢ (∀𝑧 ∈ 𝐴 Tr 𝑧 → E Po 𝐴)
Theorem	dford5reg 35835	Given ax-reg 9488, an ordinal is a transitive class totally ordered by the membership relation. (Contributed by Scott Fenton, 28-Jan-2011.)
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))
Theorem	dfon2lem1 35836	Lemma for dfon2 35845. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}
Theorem	dfon2lem2 35837*	Lemma for dfon2 35845. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴
Theorem	dfon2lem3 35838*	Lemma for dfon2 35845. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧)))
Theorem	dfon2lem4 35839*	Lemma for dfon2 35845. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
Theorem	dfon2lem5 35840*	Lemma for dfon2 35845. Two sets satisfying the new definition also satisfy trichotomy with respect to ∈. (Contributed by Scott Fenton, 25-Feb-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
Theorem	dfon2lem6 35841*	Lemma for dfon2 35845. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)
⊢ ((Tr 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑧((𝑧 ⊊ 𝑥 ∧ Tr 𝑧) → 𝑧 ∈ 𝑥)) → ∀𝑦((𝑦 ⊊ 𝑆 ∧ Tr 𝑦) → 𝑦 ∈ 𝑆))
Theorem	dfon2lem7 35842*	Lemma for dfon2 35845. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)))
Theorem	dfon2lem8 35843*	Lemma for dfon2 35845. The intersection of a nonempty class 𝐴 of new ordinals is itself a new ordinal and is contained within 𝐴 (Contributed by Scott Fenton, 26-Feb-2011.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)) → (∀𝑧((𝑧 ⊊ ∩ 𝐴 ∧ Tr 𝑧) → 𝑧 ∈ ∩ 𝐴) ∧ ∩ 𝐴 ∈ 𝐴))
Theorem	dfon2lem9 35844*	Lemma for dfon2 35845. A class of new ordinals is well-founded by E. (Contributed by Scott Fenton, 3-Mar-2011.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → E Fr 𝐴)
Theorem	dfon2 35845*	On consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers", American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.)
⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
Theorem	rdgprc0 35846	The value of the recursive definition generator at ∅ when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
Theorem	rdgprc 35847	The value of the recursive definition generator when 𝐼 is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))
Theorem	dfrdg2 35848*	Alternate definition of the recursive function generator when 𝐼 is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝐼 ∈ 𝑉 → rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
Theorem	dfrdg3 35849*	Generalization of dfrdg2 35848 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
21.11.12  Defined equality axioms
Theorem	axextdfeq 35850	A version of ax-ext 2705 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)))
Theorem	ax8dfeq 35851	A version of ax-8 2115 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦))
Theorem	axextdist 35852	ax-ext 2705 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
Theorem	axextbdist 35853	axextb 2708 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
Theorem	19.12b 35854*	Version of 19.12vv 2349 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
Theorem	exnel 35855	There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦
Theorem	distel 35856	Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5384 and elirrv 9493.) (Contributed by Scott Fenton, 15-Dec-2010.)
⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)
Theorem	axextndbi 35857	axextnd 10492 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
⊢ ∃𝑧(𝑥 = 𝑦 ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
21.11.13  Hypothesis builders
Theorem	hbntg 35858	A more general form of hbnt 2298. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
Theorem	hbimtg 35859	A more general and closed form of hbim 2303. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒 → 𝜓) → ∀𝑥(𝜑 → 𝜃)))
Theorem	hbaltg 35860	A more general and closed form of hbal 2172. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜓))
Theorem	hbng 35861	A more general form of hbn 2299. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)
Theorem	hbimg 35862	A more general form of hbim 2303. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜒 → ∀𝑥𝜃)    ⇒   ⊢ ((𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜃))
21.11.14  Well-founded zero, successor, and limits
Syntax	cwsuc 35863	Declare the syntax for well-founded successor.
class wsuc(𝑅, 𝐴, 𝑋)
Syntax	cwlim 35864	Declare the syntax for well-founded limit class.
class WLim(𝑅, 𝐴)
Definition	df-wsuc 35865	Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
Definition	df-wlim 35866*	Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
Theorem	wsuceq123 35867	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))
Theorem	wsuceq1 35868	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 = 𝑆 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐴, 𝑋))
Theorem	wsuceq2 35869	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋))
Theorem	wsuceq3 35870	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌))
Theorem	nfwsuc 35871	Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝑋    ⇒   ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋)
Theorem	wlimeq12 35872	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))
Theorem	wlimeq1 35873	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
⊢ (𝑅 = 𝑆 → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐴))
Theorem	wlimeq2 35874	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
⊢ (𝐴 = 𝐵 → WLim(𝑅, 𝐴) = WLim(𝑅, 𝐵))
Theorem	nfwlim 35875	Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥WLim(𝑅, 𝐴)
Theorem	elwlim 35876	Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))
Theorem	wzel 35877	The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐴 ≠ ∅) → inf(𝐴, 𝐴, 𝑅) ∈ 𝐴)
Theorem	wsuclem 35878*	Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)))
Theorem	wsucex 35879	Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)
Theorem	wsuccl 35880*	If 𝑋 is a set with an 𝑅 successor in 𝐴, then its well-founded successor is a member of 𝐴. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)    ⇒   ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)
Theorem	wsuclb 35881	A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑅𝑌)    ⇒   ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))
Theorem	wlimss 35882	The class of limit points is a subclass of the base class. (Contributed by Scott Fenton, 16-Jun-2018.)
⊢ WLim(𝑅, 𝐴) ⊆ 𝐴
21.11.15  Quantifier-free definitions
Syntax	ctxp 35883	Declare the syntax for tail Cartesian product.
class (𝐴 ⊗ 𝐵)
Syntax	cpprod 35884	Declare the syntax for the parallel product.
class pprod(𝑅, 𝑆)
Syntax	csset 35885	Declare the subset relationship class.
class SSet
Syntax	ctrans 35886	Declare the transitive set class.
class Trans
Syntax	cbigcup 35887	Declare the set union relationship.
class Bigcup
Syntax	cfix 35888	Declare the syntax for the fixpoints of a class.
class Fix 𝐴
Syntax	climits 35889	Declare the class of limit ordinals.
class Limits
Syntax	cfuns 35890	Declare the syntax for the class of all function.
class Funs
Syntax	csingle 35891	Declare the syntax for the singleton function.
class Singleton
Syntax	csingles 35892	Declare the syntax for the class of all singletons.
class Singletons
Syntax	cimage 35893	Declare the syntax for the image functor.
class Image𝐴
Syntax	ccart 35894	Declare the syntax for the cartesian function.
class Cart
Syntax	cimg 35895	Declare the syntax for the image function.
class Img
Syntax	cdomain 35896	Declare the syntax for the domain function.
class Domain
Syntax	crange 35897	Declare the syntax for the range function.
class Range
Syntax	capply 35898	Declare the syntax for the application function.
class Apply
Syntax	ccup 35899	Declare the syntax for the cup function.
class Cup
Syntax	ccap 35900	Declare the syntax for the cap function.
class Cap