P. 382 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38101-38200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tsan1 38101	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
Theorem	tsan2 38102	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ ¬ (𝜑 ∧ 𝜓)))
Theorem	tsan3 38103	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))
Theorem	tsna1 38104	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)))
Theorem	tsna2 38105	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ (𝜑 ⊼ 𝜓)))
Theorem	tsna3 38106	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜓 ∨ (𝜑 ⊼ 𝜓)))
Theorem	tsor1 38107	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
Theorem	tsor2 38108	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜑 ∨ (𝜑 ∨ 𝜓)))
Theorem	tsor3 38109	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 ∨ 𝜓)))
Theorem	ts3an1 38110	A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ∨ (𝜑 ∧ 𝜓 ∧ 𝜒)))
Theorem	ts3an2 38111	A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)))
Theorem	ts3an3 38112	A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (𝜒 ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)))
Theorem	ts3or1 38113	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (((𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ 𝜓 ∨ 𝜒)))
Theorem	ts3or2 38114	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜓 ∨ 𝜒)))
Theorem	ts3or3 38115	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜒 ∨ (𝜑 ∨ 𝜓 ∨ 𝜒)))
21.25.3  Equality deductions A collection of theorems for commuting equalities (or biconditionals) with other constructs.
Theorem	iuneq2f 38116	Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	rabeq12f 38117	Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	csbeq12 38118	Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷)
Theorem	sbeqi 38119	Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ ((𝑥 = 𝑦 ∧ ∀𝑧(𝜑 ↔ 𝜓)) → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜓))
Theorem	ralbi12f 38120	Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	oprabbi 38121	Equality deduction for class abstraction of nested ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
Theorem	mpobi123f 38122*	Equality deduction for maps-to notations with two arguments. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑦𝐷    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ Ⅎ𝑥𝐷    ⇒   ⊢ (((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝐸 = 𝐹) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹))
Theorem	iuneq12f 38123	Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
Theorem	iineq12f 38124	Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)
Theorem	opabbi 38125	Equality deduction for class abstraction of ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Theorem	mptbi12f 38126	Equality deduction for maps-to notations. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → (𝑥 ∈ 𝐴 ↦ 𝐷) = (𝑥 ∈ 𝐵 ↦ 𝐸))
21.25.4  Miscellanea Work in progress or things that do not belong anywhere else.
Theorem	orcomdd 38127	Commutativity of logic disjunction, in double deduction form. Should not be moved to main, see PR #3034 in Github. Use orcomd 870 instead. (Contributed by Giovanni Mascellani, 19-Mar-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 ∨ 𝜒)))
Theorem	scottexf 38128*	A version of scottex 9954 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Theorem	scott0f 38129*	A version of scott0 9955 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Theorem	scottn0f 38130*	A version of scott0f 38129 with inequalities instead of equalities. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≠ ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
Theorem	ac6s3f 38131*	Generalization of the Axiom of Choice to classes, with bound-variable hypothesis. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 𝜓)
Theorem	ac6s6 38132*	Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)
Theorem	ac6s6f 38133*	Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 20-Aug-2018.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)
21.26  Mathbox for Peter Mazsa
21.26.1  Notations
Syntax	cxrn 38134	Extend the definition of a class to include the range Cartesian product class.
class (𝐴 ⋉ 𝐵)
Syntax	ccoss 38135	Extend the definition of a class to include the class of cosets by a class. (Read: the class of cosets by 𝑅.)
class ≀ 𝑅
Syntax	ccoels 38136	Extend the definition of a class to include the class of coelements on a class. (Read: the class of coelements on 𝐴.)
class ∼ 𝐴
Syntax	crels 38137	Extend the definition of a class to include the relation class.
class Rels
Syntax	cssr 38138	Extend the definition of a class to include the subset class.
class S
Syntax	crefs 38139	Extend the definition of a class to include the reflexivity class.
class Refs
Syntax	crefrels 38140	Extend the definition of a class to include the reflexive relations class.
class RefRels
Syntax	wrefrel 38141	Extend the definition of a wff to include the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.)
wff RefRel 𝑅
Syntax	ccnvrefs 38142	Extend the definition of a class to include the converse reflexivity class.
class CnvRefs
Syntax	ccnvrefrels 38143	Extend the definition of a class to include the converse reflexive relations class.
class CnvRefRels
Syntax	wcnvrefrel 38144	Extend the definition of a wff to include the converse reflexive relation predicate. (Read: 𝑅 is a converse reflexive relation.)
wff CnvRefRel 𝑅
Syntax	csyms 38145	Extend the definition of a class to include the symmetry class.
class Syms
Syntax	csymrels 38146	Extend the definition of a class to include the symmetry relations class.
class SymRels
Syntax	wsymrel 38147	Extend the definition of a wff to include the symmetry relation predicate. (Read: 𝑅 is a symmetric relation.)
wff SymRel 𝑅
Syntax	ctrs 38148	Extend the definition of a class to include the transitivity class (but cf. the transitive class defined in df-tr 5284).
class Trs
Syntax	ctrrels 38149	Extend the definition of a class to include the transitive relations class.
class TrRels
Syntax	wtrrel 38150	Extend the definition of a wff to include the transitive relation predicate. (Read: 𝑅 is a transitive relation.)
wff TrRel 𝑅
Syntax	ceqvrels 38151	Extend the definition of a class to include the equivalence relations class.
class EqvRels
Syntax	weqvrel 38152	Extend the definition of a wff to include the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.)
wff EqvRel 𝑅
Syntax	ccoeleqvrels 38153	Extend the definition of a class to include the coelement equivalence relations class.
class CoElEqvRels
Syntax	wcoeleqvrel 38154	Extend the definition of a wff to include the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.)
wff CoElEqvRel 𝐴
Syntax	credunds 38155	Extend the definition of a class to include the redundancy class.
class Redunds
Syntax	wredund 38156	Extend the definition of a wff to include the redundancy predicate. (Read: 𝐴 is redundant with respect to 𝐵 in 𝐶.)
wff 𝐴 Redund ⟨𝐵, 𝐶⟩
Syntax	wredundp 38157	Extend wff definition to include the redundancy operator for propositions.
wff redund (𝜑, 𝜓, 𝜒)
Syntax	cdmqss 38158	Extend the definition of a class to include the domain quotients class.
class DomainQss
Syntax	wdmqs 38159	Extend the definition of a wff to include the domain quotient predicate. (Read: the domain quotient of 𝑅 is 𝐴.)
wff 𝑅 DomainQs 𝐴
Syntax	cers 38160	Extend the definition of a class to include the equivalence relations on their domain quotients class.
class Ers
Syntax	werALTV 38161	Extend the definition of a wff to include the equivalence relation on its domain quotient predicate. (Read: 𝑅 is an equivalence relation on its domain quotient 𝐴.)
wff 𝑅 ErALTV 𝐴
Syntax	ccomembers 38162	Extend the definition of a class to include the comember equivalence relations class.
class CoMembErs
Syntax	wcomember 38163	Extend the definition of a wff to include the comember equivalence relation predicate. (Read: the comember equivalence relation on 𝐴, or, the restricted coelement equivalence relation on its domain quotient 𝐴.)
wff CoMembEr 𝐴
Syntax	cfunss 38164	Extend the definition of a class to include the function set class.
class Funss
Syntax	cfunsALTV 38165	Extend the definition of a class to include the functions class, i.e., the function relations class.
class FunsALTV
Syntax	wfunALTV 38166	Extend the definition of a wff to include the function predicate, i.e., the function relation predicate. (Read: 𝐹 is a function.)
wff FunALTV 𝐹
Syntax	cdisjss 38167	Extend the definition of a class to include the disjoint set class.
class Disjss
Syntax	cdisjs 38168	Extend the definition of a class to include the disjoints class, i.e., the disjoint relations class.
class Disjs
Syntax	wdisjALTV 38169	Extend the definition of a wff to include the disjoint predicate, i.e., the disjoint relation predicate. (Read: 𝑅 is a disjoint.)
wff Disj 𝑅
Syntax	celdisjs 38170	Extend the definition of a class to include the disjoint elements class, i.e., the disjoint element relations class.
class ElDisjs
Syntax	weldisj 38171	Extend the definition of a wff to include the disjoint element predicate, i.e., the disjoint element relation predicate. (Read: the elements of 𝐴 are disjoint.)
wff ElDisj 𝐴
Syntax	wantisymrel 38172	Extend the definition of a wff to include the antisymmetry relation predicate. (Read: 𝑅 is an antisymmetric relation.)
wff AntisymRel 𝑅
Syntax	cparts 38173	Extend the definition of a class to include the partitions class, i.e., the partition relations class.
class Parts
Syntax	wpart 38174	Extend the definition of a wff to include the partition predicate, i.e., the partition relation predicate. (Read: 𝐴 is a partition by 𝑅.)
wff 𝑅 Part 𝐴
Syntax	cmembparts 38175	Extend the definition of a class to include the member partitions class, i.e., the member partition relations class.
class MembParts
Syntax	wmembpart 38176	Extend the definition of a wff to include the member partition predicate, i.e., the member partition relation predicate. (Read: 𝐴 is a member partition.)
wff MembPart 𝐴
21.26.2  Preparatory theorems
Theorem	el2v1 38177	New way (elv 3493, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.)
⊢ ((𝑥 ∈ V ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	el3v1 38178	New way (elv 3493, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
⊢ ((𝑥 ∈ V ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
Theorem	el3v2 38179	New way (elv 3493, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
⊢ ((𝜑 ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	el3v12 38180	New way (elv 3493, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	el3v13 38181	New way (elv 3493, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝑥 ∈ V ∧ 𝜓 ∧ 𝑧 ∈ V) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	el3v23 38182	New way (elv 3493, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝜑 ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	anan 38183	Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜑 ∧ 𝜃) ∧ 𝜏)) ↔ ((𝜓 ∧ 𝜃) ∧ (𝜑 ∧ (𝜒 ∧ 𝜏))))
Theorem	triantru3 38184	A wff is equivalent to its conjunctions with truths. (Contributed by Peter Mazsa, 30-Nov-2018.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	bianim 38185	Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))
Theorem	biorfd 38186	A wff is equivalent to its disjunction with falsehood, deduction form. (Contributed by Peter Mazsa, 22-Aug-2023.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 ↔ (𝜓 ∨ 𝜒)))
Theorem	eqbrtr 38187	Substitution of equal classes in binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Theorem	eqbrb 38188	Substitution of equal classes in a binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶))
Theorem	eqeltr 38189	Substitution of equal classes into element relation. (Contributed by Peter Mazsa, 22-Jul-2017.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)
Theorem	eqelb 38190	Substitution of equal classes into element relation. (Contributed by Peter Mazsa, 17-Jul-2019.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶))
Theorem	eqeqan2d 38191	Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	suceqsneq 38192	One-to-one relationship between the successor operation and the singleton. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = suc 𝐵 ↔ {𝐴} = {𝐵}))
Theorem	sucdifsn2 38193	Absorption of union with a singleton by difference. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
Theorem	sucdifsn 38194	The difference between the successor and the singleton of a class is the class. (Contributed by Peter Mazsa, 20-Sep-2024.)
⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴
Theorem	disjresin 38195	The restriction to a disjoint is the empty class. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = ∅)
Theorem	disjresdisj 38196	The intersection of restrictions to disjoint is the empty class. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵)) = ∅)
Theorem	disjresdif 38197	The difference between restrictions to disjoint is the first restriction. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
Theorem	disjresundif 38198	Lemma for ressucdifsn2 38199. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
Theorem	ressucdifsn2 38199	The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 38200. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)
Theorem	ressucdifsn 38200	The difference between restrictions to the successor and the singleton of a class is the restriction to the class. (Contributed by Peter Mazsa, 20-Sep-2024.)
⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)