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Theorem List for Metamath Proof Explorer - 38801-38900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-rcl 38801* Reflexive closure of a relation. This is the smallest superset which has the reflexive property. (Contributed by RP, 5-Jun-2020.)
r* = (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})

Theoremdfrcl2 38802 Reflexive closure of a relation as union with restricted identity relation. (Contributed by RP, 6-Jun-2020.)
r* = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))

Theoremdfrcl3 38803 Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
r* = (𝑥 ∈ V ↦ ((𝑥𝑟0) ∪ (𝑥𝑟1)))

Theoremdfrcl4 38804* Reflexive closure of a relation as indexed union of powers of the relation. (Contributed by RP, 8-Jun-2020.)
r* = (𝑟 ∈ V ↦ 𝑛 ∈ {0, 1} (𝑟𝑟𝑛))

20.28.2.3  Finite relationship composition.

In order for theorems on the transitive closure of a relation to be grouped together before the concept of continuity, we really need an analogue of 𝑟 that works on finite ordinals or finite sets instead of natural numbers.

Theoremrelexp2 38805 A set operated on by the relation exponent to the second power is equal to the composition of the set with itself. (Contributed by RP, 1-Jun-2020.)
(𝑅𝑉 → (𝑅𝑟2) = (𝑅𝑅))

Theoremrelexpnul 38806 If the domain and range of powers of a relation are disjoint then the relation raised to the sum of those exponents is empty. (Contributed by RP, 1-Jun-2020.)
(((𝑅𝑉 ∧ Rel 𝑅) ∧ (𝑁 ∈ ℕ0𝑀 ∈ ℕ0)) → ((dom (𝑅𝑟𝑁) ∩ ran (𝑅𝑟𝑀)) = ∅ ↔ (𝑅𝑟(𝑁 + 𝑀)) = ∅))

Theoremeliunov2 38807* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. Generalized from dfrtrclrec2 14181. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))

Theoremeltrclrec 38808* Membership in the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅𝑟𝑛)))

Theoremelrtrclrec 38809* Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅𝑟𝑛)))

Theorembriunov2 38810* Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))

Theorembrmptiunrelexpd 38811* If two elements are connected by an indexed union of relational powers, then they are connected via 𝑛 instances the relation, for some 𝑛. Generalization of dfrtrclrec2 14181. (Contributed by RP, 21-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ⊆ ℕ0)       (𝜑 → (𝐴(𝐶𝑅)𝐵 ↔ ∃𝑛𝑁 𝐴(𝑅𝑟𝑛)𝐵))

Theoremfvmptiunrelexplb0d 38812* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → 0 ∈ 𝑁)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))

Theoremfvmptiunrelexplb0da 38813* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → Rel 𝑅)    &   (𝜑 → 0 ∈ 𝑁)       (𝜑 → ( I ↾ 𝑅) ⊆ (𝐶𝑅))

Theoremfvmptiunrelexplb1d 38814* If the indexed union ranges over the first power of the relation, then the relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → 1 ∈ 𝑁)       (𝜑𝑅 ⊆ (𝐶𝑅))

Theorembrfvid 38815 If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

TheorembrfvidRP 38816 If two elements are connected by a value of the identity relation, then they are connected via the argument. This is an example which uses brmptiunrelexpd 38811. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

Theoremfvilbd 38817 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))

TheoremfvilbdRP 38818 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))

Theorembrfvrcld 38819 If two elements are connected by the reflexive closure of a relation, then they are connected via zero or one instances the relation. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅𝑟0)𝐵𝐴(𝑅𝑟1)𝐵)))

Theorembrfvrcld2 38820 If two elements are connected by the reflexive closure of a relation, then they are equal or related by relation. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))

Theoremfvrcllb0d 38821 A restriction of the identity relation is a subset of the reflexive closure of a set. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (r*‘𝑅))

Theoremfvrcllb0da 38822 A restriction of the identity relation is a subset of the reflexive closure of a relation. (Contributed by RP, 22-Jul-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ 𝑅) ⊆ (r*‘𝑅))

Theoremfvrcllb1d 38823 A set is a subset of its image under the reflexive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (r*‘𝑅))

Theorembrtrclrec 38824* Two classes related by the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ 𝑋(𝑅𝑟𝑛)𝑌))

Theorembrrtrclrec 38825* Two classes related by the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅𝑟𝑛)𝑌))

Theorembriunov2uz 38826* Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))

Theoremeliunov2uz 38827* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))

Theoremov2ssiunov2 38828* Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14182 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))

Theoremrelexp0eq 38829 The zeroth power of relationships is the same if and only if the union of their domain and ranges is the same. (Contributed by RP, 11-Jun-2020.)
((𝐴𝑈𝐵𝑉) → ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ (𝐴𝑟0) = (𝐵𝑟0)))

Theoremiunrelexp0 38830* Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))

Theoremrelexpxpnnidm 38831 Any positive power of a cross product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
(𝑁 ∈ ℕ → ((𝐴𝑈𝐵𝑉 ∧ (𝐴𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑𝑟𝑁) = (𝐴 × 𝐵)))

Theoremrelexpiidm 38832 Any power of any restriction of the identity relation is itself. (Contributed by RP, 12-Jun-2020.)
((𝐴𝑉𝑁 ∈ ℕ0) → (( I ↾ 𝐴)↑𝑟𝑁) = ( I ↾ 𝐴))

Theoremrelexpss1d 38833 The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))

Theoremcomptiunov2i 38834* The composition two indexed unions is sometimes a similar indexed union. (Contributed by RP, 10-Jun-2020.)
𝑋 = (𝑎 ∈ V ↦ 𝑖𝐼 (𝑎 𝑖))    &   𝑌 = (𝑏 ∈ V ↦ 𝑗𝐽 (𝑏 𝑗))    &   𝑍 = (𝑐 ∈ V ↦ 𝑘𝐾 (𝑐 𝑘))    &   𝐼 ∈ V    &   𝐽 ∈ V    &   𝐾 = (𝐼𝐽)    &    𝑘𝐼 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)    &    𝑘𝐽 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)    &    𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ⊆ 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)       (𝑋𝑌) = 𝑍

Theoremcorclrcl 38835 The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(r* ∘ r*) = r*

Theoremiunrelexpmin1 38836* The indexed union of relation exponentiation over the natural numbers is the minimum transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = ℕ) → ∀𝑠((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))

Theoremrelexpmulnn 38837 With exponents limited to the counting numbers, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
(((𝑅𝑉𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Theoremrelexpmulg 38838 With ordered exponents, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
(((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) ∧ (𝐽 ∈ ℕ0𝐾 ∈ ℕ0)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Theoremtrclrelexplem 38839* The union of relational powers to positive multiples of 𝑁 is a subset to the transitive closure raised to the power of 𝑁. (Contributed by RP, 15-Jun-2020.)
(𝑁 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))

Theoremiunrelexpmin2 38840* The indexed union of relation exponentiation over the natural numbers (including zero) is the minimum reflexive-transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))

Theoremrelexp01min 38841 With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.)
(((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) ∧ (𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1})) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Theoremrelexp1idm 38842 Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.)
(𝑅𝑉 → ((𝑅𝑟1)↑𝑟1) = (𝑅𝑟1))

Theoremrelexp0idm 38843 Repeated raising a relation to the zeroth power is idempotent. (Contributed by RP, 12-Jun-2020.)
(𝑅𝑉 → ((𝑅𝑟0)↑𝑟0) = (𝑅𝑟0))

Theoremrelexp0a 38844 Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))

Theoremrelexpxpmin 38845 The composition of powers of a cross-product of non-disjoint sets is the cross product raised to the minimum exponent. (Contributed by RP, 13-Jun-2020.)
(((𝐴𝑈𝐵𝑉 ∧ (𝐴𝐵) ≠ ∅) ∧ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ 𝐽 ∈ ℕ0𝐾 ∈ ℕ0)) → (((𝐴 × 𝐵)↑𝑟𝐽)↑𝑟𝐾) = ((𝐴 × 𝐵)↑𝑟𝐼))

Theoremrelexpaddss 38846 The composition of two powers of a relation is a subset of the relation raised to the sum of those exponents. This is equality where 𝑅 is a relation as shown by relexpaddd 14178 or when the sum of the powers isn't 1 as shown by relexpaddg 14177. (Contributed by RP, 3-Jun-2020.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))

Theoremiunrelexpuztr 38847* The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 14184. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = (ℤ𝑀) ∧ 𝑀 ∈ ℕ0) → ((𝐶𝑅) ∘ (𝐶𝑅)) ⊆ (𝐶𝑅))

20.28.2.4  Transitive closure of a relation

Theoremdftrcl3 38848* Transitive closure of a relation, expressed as indexed union of powers of relations. (Contributed by RP, 5-Jun-2020.)
t+ = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))

Theorembrfvtrcld 38849* If two elements are connected by the transitive closure of a relation, then they are connected via 𝑛 instances the relation, for some counting number 𝑛. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(t+‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ 𝐴(𝑅𝑟𝑛)𝐵))

Theoremfvtrcllb1d 38850 A set is a subset of its image under the transitive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (t+‘𝑅))

Theoremtrclfvcom 38851 The transitive closure of a relation commutes with the relation. (Contributed by RP, 18-Jul-2020.)
(𝑅𝑉 → ((t+‘𝑅) ∘ 𝑅) = (𝑅 ∘ (t+‘𝑅)))

Theoremcnvtrclfv 38852 The converse of the transitive closure is equal to the transitive closure of the converse relation. (Contributed by RP, 19-Jul-2020.)
(𝑅𝑉(t+‘𝑅) = (t+‘𝑅))

Theoremcotrcltrcl 38853 The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
(t+ ∘ t+) = t+

Theoremtrclimalb2 38854 Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)
((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)

Theorembrtrclfv2 38855* Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.)
((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))

Theoremtrclfvdecomr 38856 The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.)
(𝑅𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))

Theoremtrclfvdecoml 38857 The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.)
(𝑅𝑉 → (t+‘𝑅) = (𝑅 ∪ (𝑅 ∘ (t+‘𝑅))))

TheoremdmtrclfvRP 38858 The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
(𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)

TheoremrntrclfvRP 38859 The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 19-Jul-2020.) (Proof modification is discouraged.)
(𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)

Theoremrntrclfv 38860 The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
(𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)

Theoremdfrtrcl3 38861* Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 14186. (Contributed by RP, 5-Jun-2020.)
t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))

Theorembrfvrtrcld 38862* If two elements are connected by the reflexive-transitive closure of a relation, then they are connected via 𝑛 instances the relation, for some natural number 𝑛. Similar of dfrtrclrec2 14181. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(t*‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))

Theoremfvrtrcllb0d 38863 A restriction of the identity relation is a subset of the reflexive-transitive closure of a set. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*‘𝑅))

Theoremfvrtrcllb0da 38864 A restriction of the identity relation is a subset of the reflexive-transitive closure of a relation. (Contributed by RP, 22-Jul-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ 𝑅) ⊆ (t*‘𝑅))

Theoremfvrtrcllb1d 38865 A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (t*‘𝑅))

Theoremdfrtrcl4 38866 Reflexive-transitive closure of a relation, expressed as the union of the zeroth power and the transitive closure. (Contributed by RP, 5-Jun-2020.)
t* = (𝑟 ∈ V ↦ ((𝑟𝑟0) ∪ (t+‘𝑟)))

Theoremcorcltrcl 38867 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
(r* ∘ t+) = t*

Theoremcortrcltrcl 38868 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
(t* ∘ t+) = t*

Theoremcorclrtrcl 38869 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
(r* ∘ t*) = t*

Theoremcotrclrcl 38870 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
(t+ ∘ r*) = t*

Theoremcortrclrcl 38871 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
(t* ∘ r*) = t*

Theoremcotrclrtrcl 38872 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
(t+ ∘ t*) = t*

Theoremcortrclrtrcl 38873 The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(t* ∘ t*) = t*

Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879].

Theoremfrege77d 38874 If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 77 of [Frege1879] p. 62. Compare with frege77 39069. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)    &   (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈)       (𝜑𝐵𝑈)

Theoremfrege81d 38875 If the image of 𝑈 is a subset 𝑈, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 81 of [Frege1879] p. 63. Compare with frege81 39073. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)       (𝜑𝐵𝑈)

Theoremfrege83d 38876 If the image of the union of 𝑈 and 𝑉 is a subset of the union of 𝑈 and 𝑉, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of the union of 𝑈 and 𝑉. Similar to Proposition 83 of [Frege1879] p. 65. Compare with frege83 39075. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅 “ (𝑈𝑉)) ⊆ (𝑈𝑉))       (𝜑𝐵 ∈ (𝑈𝑉))

Theoremfrege96d 38877 If 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 39088. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)

Theoremfrege87d 38878 If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 87 of [Frege1879] p. 66. Compare with frege87 39079. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶𝑅𝐵)    &   (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)       (𝜑𝐵𝑈)

Theoremfrege91d 38879 If 𝐵 follows 𝐴 in 𝑅 then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 91 of [Frege1879] p. 68. Comparw with frege91 39083. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)

Theoremfrege97d 38880 If 𝐴 contains all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 97 of [Frege1879] p. 71. Compare with frege97 39089. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 = ((t+‘𝑅) “ 𝑈))       (𝜑 → (𝑅𝐴) ⊆ 𝐴)

Theoremfrege98d 38881 If 𝐶 follows 𝐴 and 𝐵 follows 𝐶 in the transitive closure of 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 39090. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶(t+‘𝑅)𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)

Theoremfrege102d 38882 If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 39094. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)

Theoremfrege106d 38883 If 𝐵 follows 𝐴 in 𝑅, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in 𝑅. Similar to Proposition 106 of [Frege1879] p. 73. Compare with frege106 39098. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴𝑅𝐵)       (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵))

Theoremfrege108d 38884 If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 108 of [Frege1879] p. 74. Compare with frege108 39100. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))    &   (𝜑𝐶𝑅𝐵)       (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))

Theoremfrege109d 38885 If 𝐴 contains all elements of 𝑈 and all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 109 of [Frege1879] p. 74. Compare with frege109 39101. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))       (𝜑 → (𝑅𝐴) ⊆ 𝐴)

Theoremfrege114d 38886 If either 𝑅 relates 𝐴 and 𝐵 or 𝐴 and 𝐵 are the same, then either 𝐴 and 𝐵 are the same, 𝑅 relates 𝐴 and 𝐵, 𝑅 relates 𝐵 and 𝐴. Similar to Proposition 114 of [Frege1879] p. 76. Compare with frege114 39106. (Contributed by RP, 15-Jul-2020.)
(𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵))       (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵𝐵𝑅𝐴))

Theoremfrege111d 38887 If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐴 follows 𝐵 or 𝐵 and 𝐴 in the transitive closure of 𝑅. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 39103. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))    &   (𝜑𝐶𝑅𝐵)       (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵𝐵(t+‘𝑅)𝐴))

Theoremfrege122d 38888 If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 is the successor of 𝑋, then 𝐴 and 𝐵 are the same (or 𝐵 follows 𝐴 in the transitive closure of 𝐹). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 39114. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴 = (𝐹𝑋))    &   (𝜑𝐵 = (𝐹𝑋))       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))

Theoremfrege124d 38889 If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 124 of [Frege1879] p. 80. Compare with frege124 39116. (Contributed by RP, 16-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝑋 ∈ dom 𝐹)    &   (𝜑𝐴 = (𝐹𝑋))    &   (𝜑𝑋(t+‘𝐹)𝐵)    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))

Theoremfrege126d 38890 If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 126 of [Frege1879] p. 81. Compare with frege126 39118. (Contributed by RP, 16-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝑋 ∈ dom 𝐹)    &   (𝜑𝐴 = (𝐹𝑋))    &   (𝜑𝑋(t+‘𝐹)𝐵)    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵𝐵(t+‘𝐹)𝐴))

Theoremfrege129d 38891 If 𝐹 is a function and (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹, the successor of 𝐴 is either 𝐵 or it follows 𝐵 or it comes before 𝐵 in the transitive closure of 𝐹. Similar to Proposition 129 of [Frege1879] p. 83. Comparw with frege129 39121. (Contributed by RP, 16-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝐴 ∈ dom 𝐹)    &   (𝜑𝐶 = (𝐹𝐴))    &   (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵𝐵(t+‘𝐹)𝐴))    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐵(t+‘𝐹)𝐶𝐵 = 𝐶𝐶(t+‘𝐹)𝐵))

Theoremfrege131d 38892 If 𝐹 is a function and 𝐴 contains all elements of 𝑈 and all elements before or after those elements of 𝑈 in the transitive closure of 𝐹, then the image under 𝐹 of 𝐴 is a subclass of 𝐴. Similar to Proposition 131 of [Frege1879] p. 85. Compare with frege131 39123. (Contributed by RP, 17-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝐴 = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐹𝐴) ⊆ 𝐴)

Theoremfrege133d 38893 If 𝐹 is a function and 𝐴 and 𝐵 both follow 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹 (or both if it loops). Similar to Proposition 133 of [Frege1879] p. 86. Compare with frege133 39125. (Contributed by RP, 18-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝑋(t+‘𝐹)𝐴)    &   (𝜑𝑋(t+‘𝐹)𝐵)    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵𝐵(t+‘𝐹)𝐴))

20.28.3  Propositions from _Begriffsschrift_

In 1879, Frege introduced notation for documenting formal reasoning about propositions (and classes) which covered elements of propositional logic, predicate calculus and reasoning about relations. However, due to the pitfalls of naive set theory, adapting this work for inclusion in set.mm required dividing statements about propositions from those about classes and identifying when a restriction to sets is required. For an overview comparing the details of Frege's two-dimensional notation and that used in set.mm, see mmfrege.html. See ru 3661 for discussion of an example of a class that is not a set.

Numbered propositions from [Frege1879]. ax-frege1 38919, ax-frege2 38920, ax-frege8 38938, ax-frege28 38959, ax-frege31 38963, ax-frege41 38974, frege52 (see ax-frege52a 38986, frege52b 39018, and ax-frege52c 39017 for translations), frege54 (see ax-frege54a 38991, frege54b 39022 and ax-frege54c 39021 for translations) and frege58 (see ax-frege58a 39004, ax-frege58b 39030 and frege58c 39050 for translations) are considered "core" or axioms. However, at least ax-frege8 38938 can be derived from ax-frege1 38919 and ax-frege2 38920, see axfrege8 38936.

Frege introduced implication, negation and the universal quantifier as primitives and did not in the numbered propositions use other logical connectives other than equivalence introduced in ax-frege52a 38986, frege52b 39018, and ax-frege52c 39017. In dffrege69 39061, Frege introduced 𝑅 hereditary 𝐴 to say that relation 𝑅, when restricted to operate on elements of class 𝐴, will only have elements of class 𝐴 in its domain; see df-he 38902 for a definition in terms of image and subset. In dffrege76 39068, Frege introduced notation for the concept of two sets related by the transitive closure of a relation, for which we write 𝑋(t+‘𝑅)𝑌, which requires 𝑅 to also be a set. In dffrege99 39091, Frege introduced notation for the concept of two sets either identical or related by the transitive closure of a relation, for which we write 𝑋((t+‘𝑅) ∪ I )𝑌, which is a superclass of sets related by the reflexive-transitive relation 𝑋(t*‘𝑅)𝑌. Finally, in dffrege115 39107, Frege introduced notation for the concept of a relation having the property elements in its domain pair up with only one element each in its range, for which we write Fun 𝑅 (to ignore any non-relational content of the class 𝑅). Frege did this without the expressing concept of a relation (or its transitive closure) as a class, and needed to invent conventions for discussing indeterminate propositions with two slots free and how to recognize which of the slots was domain and which was range. See mmfrege.html for details.

English translations for specific propositions lifted in part from a translation by Stefan Bauer-Mengelberg as reprinted in From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. An attempt to align these propositions in the larger set.mm database has also been made. See frege77d 38874 for an example.

20.28.3.1  _Begriffsschrift_ Chapter I

Section 2 introduces the turnstile which turns an idea which may be true 𝜑 into an assertion that it does hold true 𝜑. Section 5 introduces implication, (𝜑𝜓). Section 6 introduces the single rule of interference relied upon, modus ponens ax-mp 5. Section 7 introduces negation and with in synonyms for or 𝜑𝜓), and ¬ (𝜑 → ¬ 𝜓), and two for exclusive-or corresponding to df-or 879, df-an 387, dfxor4 38894, dfxor5 38895.

Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biimplication (𝜑𝜓) or class equality 𝐴 = 𝐵 in this adaptation. Section 10 introduces "truth functions" for one or two variables in equally troubling notation, as the arguments may be understood to be logical predicates or collections. Here f(𝜑) is interpreted to mean if-(𝜑, 𝜓, 𝜒) where the content of the "function" is specified by the latter two argments or logical equivalent, while g(𝐴) is read as 𝐴𝐺 and h(𝐴, 𝐵) as 𝐴𝐻𝐵. This necessarily introduces a need for set theory as both 𝐴𝐺 and 𝐴𝐻𝐵 cannot hold unless 𝐴 is a set. (Also 𝐵.)

Section 11 introduces notation for generality, but there is no standard notation for generality when the variable is a proposition because it was realized after Frege that the universe of all possible propositions includes paradoxical constructions leading to the failure of naive set theory. So adopting f(𝜑) as if-(𝜑, 𝜓, 𝜒) would result in the translation of 𝜑 f (𝜑) as (𝜓𝜒). For collections, we must generalize over set variables or run into the same problems; this leads to 𝐴 g(𝐴) being translated as 𝑎𝑎𝐺 and so forth.

Under this interpreation the text of section 11 gives us sp 2224 (or simpl 476 and simpr 479 and anifp 1097 in the propositional case) and statements similar to cbvalivw 2111, ax-gen 1894, alrimiv 2026, and alrimdv 2028. These last four introduce a generality and have no useful definition in terms of propositional variables.

Section 12 introduces some combinations of primitive symbols and their human language counterparts. Using class notation, these can also be expressed without dummy variables. All are A, 𝑥𝑥𝐴, ¬ ∃𝑥¬ 𝑥𝐴 alex 1924, 𝐴 = V eqv 3420; Some are not B, ¬ ∀𝑥𝑥𝐵, 𝑥¬ 𝑥𝐵 exnal 1925, 𝐵 ⊊ V pssv 4242, 𝐵 ≠ V nev 38898; There are no C, 𝑥¬ 𝑥𝐶, ¬ ∃𝑥𝑥𝐶 alnex 1880, 𝐶 = ∅ eq0 4160; There exist D, ¬ ∀𝑥¬ 𝑥𝐷, 𝑥𝑥𝐷 df-ex 1879, ∅ ⊊ 𝐷 0pss 4240, 𝐷 ≠ ∅ n0 4162.

Notation for relations between expressions also can be written in various ways. All E are P, 𝑥(𝑥𝐸𝑥𝑃), ¬ ∃𝑥(𝑥𝐸 ∧ ¬ 𝑥𝑃) dfss6 3817, 𝐸 = (𝐸𝑃) df-ss 3812, 𝐸𝑃 dfss2 3815; No F are P, 𝑥(𝑥𝐹 → ¬ 𝑥𝑃), ¬ ∃𝑥(𝑥𝐹𝑥𝑃) alinexa 1943, (𝐹𝑃) = ∅ disj1 4245; Some G are not P, ¬ ∀𝑥(𝑥𝐺𝑥𝑃), 𝑥(𝑥𝐺 ∧ ¬ 𝑥𝑃) exanali 1959, (𝐺𝑃) ⊊ 𝐺 nssinpss 4088, ¬ 𝐺𝑃 nss 3888; Some H are P, ¬ ∀𝑥(𝑥𝐻 → ¬ 𝑥𝑃), 𝑥(𝑥𝐻𝑥𝑃) bj-exnalimn 33134, ∅ ⊊ (𝐻𝑃) 0pssin 38899, (𝐻𝑃) ≠ ∅ ndisj 4177.

Theoremdfxor4 38894 Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) ↔ ¬ ((¬ 𝜑𝜓) → ¬ (𝜑 → ¬ 𝜓)))

Theoremdfxor5 38895 Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) ↔ ¬ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑𝜓)))

Theoremdf3or2 38896 Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.)
((𝜑𝜓𝜒) ↔ (¬ 𝜑 → (¬ 𝜓𝜒)))

Theoremdf3an2 38897 Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.)
((𝜑𝜓𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))

Theoremnev 38898* Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
(𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥𝐴)

Theorem0pssin 38899* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
(∅ ⊊ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))

20.28.3.2  _Begriffsschrift_ Notation hints

The statement 𝑅 hereditary 𝐴 means relation 𝑅 is hereditary (in the sense of Frege) in the class 𝐴 or (𝑅𝐴) ⊆ 𝐴. The former is only a slight reduction in the number of symbols, but this reduces the number of floating hypotheses needed to be checked.

As Frege was not using the language of classes or sets, this naturally differs from the set-theoretic notion that a set is hereditary in a property: that all of its elements have a property and all of their elements have the property and so-on.

Theoremrp-imass 38900 If the 𝑅-image of a class 𝐴 is a subclass of 𝐵, then the restriction of 𝑅 to 𝐴 is a subset of the Cartesian product of 𝐴 and 𝐵. (Contributed by Richard Penner, 24-Dec-2019.)
((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))

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