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Theorem List for Metamath Proof Explorer - 39701-39800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremitgpowd 39701* The integral of a monomial on a closed bounded interval of the real line. Co-authors TA and MC. (Contributed by Jon Pennant, 31-May-2019.) (Revised by Thierry Arnoux, 14-Jun-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ∫(𝐴[,]𝐵)(𝑥𝑁) d𝑥 = (((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1))) / (𝑁 + 1)))
 
Theoremarearect 39702 The area of a rectangle whose sides are parallel to the coordinate axes in (ℝ × ℝ) is its width multiplied by its height. (Contributed by Jon Pennant, 19-Mar-2019.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   𝐷 ∈ ℝ    &   𝐴𝐵    &   𝐶𝐷    &   𝑆 = ((𝐴[,]𝐵) × (𝐶[,]𝐷))       (area‘𝑆) = ((𝐵𝐴) · (𝐷𝐶))
 
Theoremareaquad 39703* The area of a quadrilateral with two sides which are parallel to the y-axis in (ℝ × ℝ) is its width multiplied by the average height of its higher edge minus the average height of its lower edge. Co-author TA. (Contributed by Jon Pennant, 31-May-2019.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   𝐷 ∈ ℝ    &   𝐸 ∈ ℝ    &   𝐹 ∈ ℝ    &   𝐴 < 𝐵    &   𝐶𝐸    &   𝐷𝐹    &   𝑈 = (𝐶 + (((𝑥𝐴) / (𝐵𝐴)) · (𝐷𝐶)))    &   𝑉 = (𝐸 + (((𝑥𝐴) / (𝐵𝐴)) · (𝐹𝐸)))    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝑈[,]𝑉))}       (area‘𝑆) = ((((𝐹 + 𝐸) / 2) − ((𝐷 + 𝐶) / 2)) · (𝐵𝐴))
 
20.30  Mathbox for Richard Penner
 
20.30.1  Short Studies
 
20.30.1.1  Additional work on conditional logical operator
 
Theoremifpan123g 39704 Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑𝜒) ∧ (𝜑𝜏)) ∧ ((¬ 𝜓𝜃) ∧ (𝜓𝜂))))
 
Theoremifpan23 39705 Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓𝜃), (𝜒𝜏)))
 
Theoremifpdfor2 39706 Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜑, 𝜓))
 
Theoremifporcor 39707 Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)
(if-(𝜑, 𝜑, 𝜓) ↔ if-(𝜓, 𝜓, 𝜑))
 
Theoremifpdfan2 39708 Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, 𝜑))
 
Theoremifpancor 39709 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))
 
Theoremifpdfor 39710 Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, ⊤, 𝜓))
 
Theoremifpdfan 39711 Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ⊥))
 
Theoremifpbi2 39712 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))
 
Theoremifpbi3 39713 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))
 
Theoremifpim1 39714 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))
 
Theoremifpnot 39715 Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
𝜑 ↔ if-(𝜑, ⊥, ⊤))
 
Theoremifpid2 39716 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
(𝜑 ↔ if-(𝜑, ⊤, ⊥))
 
Theoremifpim2 39717 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))
 
Theoremifpbi23 39718 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))
 
Theoremifpdfbi 39719 Define biimplication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))
 
Theoremifpbiidcor 39720 Restatement of biid 262. (Contributed by RP, 25-Apr-2020.)
if-(𝜑, 𝜑, ¬ 𝜑)
 
Theoremifpbicor 39721 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
 
Theoremifpxorcor 39722 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑))
 
Theoremifpbi1 39723 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)))
 
Theoremifpnot23 39724 Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
(¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))
 
Theoremifpnotnotb 39725 Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ¬ if-(𝜑, 𝜓, 𝜒))
 
Theoremifpnorcor 39726 Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜑, ¬ 𝜓) ↔ if-(𝜓, ¬ 𝜓, ¬ 𝜑))
 
Theoremifpnancor 39727 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜓, ¬ 𝜑) ↔ if-(𝜓, ¬ 𝜑, ¬ 𝜓))
 
Theoremifpnot23b 39728 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒))
 
Theoremifpbiidcor2 39729 Restatement of biid 262. (Contributed by RP, 25-Apr-2020.)
¬ if-(𝜑, ¬ 𝜑, 𝜑)
 
Theoremifpnot23c 39730 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, 𝜒))
 
Theoremifpnot23d 39731 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒))
 
Theoremifpdfnan 39732 Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, ⊤))
 
Theoremifpdfxor 39733 Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜓))
 
Theoremifpbi12 39734 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜏)))
 
Theoremifpbi13 39735 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃)))
 
Theoremifpbi123 39736 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))
 
Theoremifpidg 39737 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜃 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑𝜓) → 𝜃) ∧ ((𝜑𝜃) → 𝜓)) ∧ ((𝜒 → (𝜑𝜃)) ∧ (𝜃 → (𝜑𝜒)))))
 
Theoremifpid3g 39738 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑𝜓) → 𝜒) ∧ ((𝜑𝜒) → 𝜓)))
 
Theoremifpid2g 39739 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜓 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜓 → (𝜑𝜒)) ∧ (𝜒 → (𝜑𝜓))))
 
Theoremifpid1g 39740 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜒𝜑) ∧ (𝜑𝜓)))
 
Theoremifpim23g 39741 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(((𝜑𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ (((𝜑𝜓) → 𝜒) ∧ (𝜒 → (𝜑𝜓))))
 
Theoremifpim3 39742 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑))
 
Theoremifpnim1 39743 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(¬ (𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜑))
 
Theoremifpim4 39744 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑))
 
Theoremifpnim2 39745 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(¬ (𝜑𝜓) ↔ if-(𝜓, ¬ 𝜓, 𝜑))
 
Theoremifpim123g 39746 Implication of conditional logical operators. The right hand side is basically conjunctive normal form which is useful in proofs. (Contributed by RP, 16-Apr-2020.)
((if-(𝜑, 𝜒, 𝜏) → if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (𝜏𝜂)))))
 
Theoremifpim1g 39747 Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ (((𝜓𝜑) ∨ (𝜃𝜒)) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))
 
Theoremifp1bi 39748 Substitute the first element of conditional logical operator. (Contributed by RP, 20-Apr-2020.)
((if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)) ↔ ((((𝜑𝜓) ∨ (𝜒𝜃)) ∧ ((𝜑𝜓) ∨ (𝜃𝜒))) ∧ (((𝜓𝜑) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜃𝜒)))))
 
Theoremifpbi1b 39749 When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜒, 𝜒) ↔ if-(𝜓, 𝜒, 𝜒))
 
Theoremifpimimb 39750 Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
 
Theoremifpororb 39751 Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∨ if-(𝜑, 𝜒, 𝜏)))
 
Theoremifpananb 39752 Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∧ if-(𝜑, 𝜒, 𝜏)))
 
Theoremifpnannanb 39753 Factor conditional logic operator over nand in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ⊼ if-(𝜑, 𝜒, 𝜏)))
 
Theoremifpor123g 39754 Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (𝜏𝜂)))))
 
Theoremifpimim 39755 Consequnce of implication. (Contributed by RP, 17-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
 
Theoremifpbibib 39756 Factor conditional logic operator over biimplication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏)))
 
Theoremifpxorxorb 39757 Factor conditional logic operator over xor in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ⊻ if-(𝜑, 𝜒, 𝜏)))
 
20.30.1.2  Sophisms
 
Theoremrp-fakeimass 39758 A special case where implication appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)
((𝜑𝜒) ↔ (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
 
Theoremrp-fakeanorass 39759 A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.)
((𝜒𝜑) ↔ (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∧ (𝜓𝜒))))
 
Theoremrp-fakeoranass 39760 A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)
((𝜑𝜒) ↔ (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒))))
 
Theoremrp-fakeinunass 39761 A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.)
(𝐶𝐴 ↔ ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)))
 
Theoremrp-fakeuninass 39762 A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)
(𝐴𝐶 ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
 
20.30.1.3  Finite Sets

Membership in the class of finite sets can be expressed in many ways.

 
Theoremrp-isfinite5 39763* A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ0. (Contributed by RP, 3-Mar-2020.)
(𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
 
Theoremrp-isfinite6 39764* A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ. (Contributed by RP, 10-Mar-2020.)
(𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
 
20.30.1.4  General Observations
 
Theoremintabssd 39765* When for each element 𝑦 there is a subset 𝐴 which may substituted for 𝑥 such that 𝑦 satisfying 𝜒 implies 𝑥 satisfies 𝜓 then the intersection of all 𝑥 that satisfy 𝜓 is a subclass the intersection of all 𝑦 that satisfy 𝜒. (Contributed by RP, 17-Oct-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))    &   (𝜑𝐴𝑦)       (𝜑 {𝑥𝜓} ⊆ {𝑦𝜒})
 
Theoremeu0 39766* There is only one empty set. (Contributed by RP, 1-Oct-2023.)
(∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥)
 
Theoremepelon2 39767 Over the ordinal numbers, one may define the relation 𝐴 E 𝐵 iff 𝐴𝐵 and one finds that, under this ordering, On is a well-ordered class, see epweon 7485. This is a weak form of epelg 5460 which only requires that we know 𝐵 to be a set. (Contributed by RP, 27-Sep-2023.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵𝐴𝐵))
 
Theoremontric3g 39768* For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥𝑦, 𝑦 = 𝑥, or 𝑦𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.)
𝑥 ∈ On ∀𝑦 ∈ On ((𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥𝑦𝑦𝑥)) ∧ (𝑦𝑥 ↔ ¬ (𝑥𝑦𝑦 = 𝑥)))
 
Theoremdfsucon 39769* 𝐴 is called a successor ordinal if it is not a limit ordinal and not the empty set. (Contributed by RP, 11-Nov-2023.)
((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
 
Theoremsnen1g 39770 A singleton is equinumerous to ordinal one iff its content is a set. (Contributed by RP, 8-Oct-2023.)
({𝐴} ≈ 1o𝐴 ∈ V)
 
Theoremsnen1el 39771 A singleton is equinumerous to ordinal one if its content is an element of it. (Contributed by RP, 8-Oct-2023.)
({𝐴} ≈ 1o𝐴 ∈ {𝐴})
 
Theoremsn1dom 39772 A singleton is dominated by ordinal one. (Contributed by RP, 29-Oct-2023.)
{𝐴} ≼ 1o
 
Theorempr2dom 39773 An unordered pair is dominated by ordinal two. (Contributed by RP, 29-Oct-2023.)
{𝐴, 𝐵} ≼ 2o
 
Theoremtr3dom 39774 An unordered triple is dominated by ordinal three. (Contributed by RP, 29-Oct-2023.)
{𝐴, 𝐵, 𝐶} ≼ 3o
 
Theoremensucne0 39775 A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof shortened by SN, 16-Nov-2023.)
(𝐴 ≈ suc 𝐵𝐴 ≠ ∅)
 
Theoremensucne0OLD 39776 A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≈ suc 𝐵𝐴 ≠ ∅)
 
Theoremnndomog 39777 Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo 8701 when both are natural numbers. (Originally by NM, 17-Jun-1998.) (Contributed by RP, 5-Nov-2023.)
((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
 
Theoremdfom6 39778 Let ω be defined to be the union of the set of all finite ordinals. (Contributed by RP, 27-Sep-2023.)
ω = (On ∩ Fin)
 
Theoreminfordmin 39779 ω is the smallest infinite ordinal. (Contributed by RP, 27-Sep-2023.)
𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥
 
Theoremiscard4 39780 Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)
((card‘𝐴) = 𝐴𝐴 ∈ ran card)
 
Theoremiscard5 39781* Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)
((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 ¬ 𝑥𝐴))
 
Theoremelrncard 39782* Let us define a cardinal number to be an element 𝐴 ∈ On such that 𝐴 is not equipotent with any 𝑥𝐴. (Contributed by RP, 1-Oct-2023.)
(𝐴 ∈ ran card ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 ¬ 𝑥𝐴))
 
Theoremharsucnn 39783 The next cardinal after a finite ordinal is the successor ordinal. (Contributed by RP, 5-Nov-2023.)
(𝐴 ∈ ω → (har‘𝐴) = suc 𝐴)
 
Theoremharval3 39784* (har‘𝐴) is the least cardinal that is greater than 𝐴. (Contributed by RP, 4-Nov-2023.)
(𝐴 ∈ dom card → (har‘𝐴) = {𝑥 ∈ ran card ∣ 𝐴𝑥})
 
Theoremharval3on 39785* For any ordinal number 𝐴 let (har‘𝐴) denote the least cardinal that is greater than 𝐴; (Contributed by RP, 4-Nov-2023.)
(𝐴 ∈ On → (har‘𝐴) = {𝑥 ∈ ran card ∣ 𝐴𝑥})
 
Theoremen2pr 39786* A class is equinumerous to ordinal two iff it is a pair of distinct sets. (Contributed by RP, 11-Oct-2023.)
(𝐴 ≈ 2o ↔ ∃𝑥𝑦(𝐴 = {𝑥, 𝑦} ∧ 𝑥𝑦))
 
Theorempr2cv 39787 If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.)
({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theorempr2el1 39788 If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐴 ∈ {𝐴, 𝐵})
 
Theorempr2cv1 39789 If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐴 ∈ V)
 
Theorempr2el2 39790 If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐵 ∈ {𝐴, 𝐵})
 
Theorempr2cv2 39791 If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐵 ∈ V)
 
Theorempren2 39792 An unordered pair is equinumerous to ordinal two iff both parts are sets not equal to each other. (Contributed by RP, 8-Oct-2023.)
({𝐴, 𝐵} ≈ 2o ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐵))
 
Theorempr2eldif1 39793 If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐴 ∈ ({𝐴, 𝐵} ∖ {𝐵}))
 
Theorempr2eldif2 39794 If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐵 ∈ ({𝐴, 𝐵} ∖ {𝐴}))
 
Theorempren2d 39795 A pair of two distinct sets is equinumerous to ordinal two. (Contributed by RP, 21-Oct-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐴𝐵)       (𝜑 → {𝐴, 𝐵} ≈ 2o)
 
Theoremaleph1min 39796 (ℵ‘1o) is the least uncountable ordinal. (Contributed by RP, 18-Nov-2023.)
(ℵ‘1o) = {𝑥 ∈ On ∣ ω ≺ 𝑥}
 
Theoremalephiso2 39797 is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)
ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})
 
Theoremalephiso3 39798 is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)
ℵ Isom E , ≺ (On, (ran card ∖ ω))
 
20.30.1.5  Infinite Sets
 
Theorempwelg 39799* The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.)
(∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
 
Theorempwinfig 39800* The powerclass of an infinite set is an infinite set, and vice-versa. Here 𝐵 is a class which is closed under both the union and the powerclass operations and which may have infinite sets as members. (Contributed by RP, 21-Mar-2020.)
(∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴 ∈ (𝐵 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝐵 ∖ Fin)))
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