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Theorem List for Metamath Proof Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfacacn 9001 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
(CHOICE ↔ ∀𝑥AC 𝑥 = V)
 
Theoremdfac13 9002 The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.)
(CHOICE ↔ ∀𝑥 𝑥AC 𝑥)
 
Theoremdfac12lem1 9003* Lemma for dfac12 9009. (Contributed by Mario Carneiro, 29-May-2015.)
(𝜑𝐴 ∈ On)    &   (𝜑𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)    &   𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))    &   (𝜑𝐶 ∈ On)    &   𝐻 = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶))       (𝜑 → (𝐺𝐶) = (𝑦 ∈ (𝑅1𝐶) ↦ if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦)))))
 
Theoremdfac12lem2 9004* Lemma for dfac12 9009. (Contributed by Mario Carneiro, 29-May-2015.)
(𝜑𝐴 ∈ On)    &   (𝜑𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)    &   𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))    &   (𝜑𝐶 ∈ On)    &   𝐻 = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∀𝑧𝐶 (𝐺𝑧):(𝑅1𝑧)–1-1→On)       (𝜑 → (𝐺𝐶):(𝑅1𝐶)–1-1→On)
 
Theoremdfac12lem3 9005* Lemma for dfac12 9009. (Contributed by Mario Carneiro, 29-May-2015.)
(𝜑𝐴 ∈ On)    &   (𝜑𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)    &   𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))       (𝜑 → (𝑅1𝐴) ∈ dom card)
 
Theoremdfac12r 9006 The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 9009 does not assume the Axiom of Regularity. (Contributed by Mario Carneiro, 29-May-2015.)
(∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ (𝑅1 “ On) ⊆ dom card)
 
Theoremdfac12k 9007* Equivalence of dfac12 9009 and dfac12a 9008, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.)
(∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)
 
Theoremdfac12a 9008 The axiom of choice holds iff every ordinal has a well-orderable powerset. (Contributed by Mario Carneiro, 29-May-2015.)
(CHOICE ↔ ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
 
Theoremdfac12 9009 The axiom of choice holds iff every aleph has a well-orderable powerset. (Contributed by Mario Carneiro, 21-May-2015.)
(CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
 
Theoremkmlem1 9010* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, 1 => 2. (Contributed by NM, 5-Apr-2004.)
(∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 𝜑) → ∃𝑦𝑧𝑥 𝜓) → ∀𝑥(∀𝑧𝑥𝑤𝑥 𝜑 → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → 𝜓)))
 
Theoremkmlem2 9011* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
(∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
 
Theoremkmlem3 9012* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. The right-hand side is part of the hypothesis of 4. (Contributed by NM, 25-Mar-2004.)
((𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ ↔ ∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
 
Theoremkmlem4 9013* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
 
Theoremkmlem5 9014* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)
 
Theoremkmlem6 9015* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)
((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝜑𝐴 = ∅)) → ∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
 
Theoremkmlem7 9016* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)
((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
 
Theoremkmlem8 9017* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 4-Apr-2004.)
((¬ ∃𝑧𝑢𝑤𝑧 𝜓 → ∃𝑦𝑧𝑢 (𝑧 ≠ ∅ → ∃!𝑤 𝑤 ∈ (𝑧𝑦))) ↔ (∃𝑧𝑢𝑤𝑧 𝜓 ∨ ∃𝑦𝑦𝑢 ∧ ∀𝑧𝑢 ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
 
Theoremkmlem9 9018* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}       𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
 
Theoremkmlem10 9019* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}       (∀(∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧 𝜑) → ∃𝑦𝑧𝐴 𝜑)
 
Theoremkmlem11 9020* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}       (𝑧𝑥 → (𝑧 𝐴) = (𝑧 (𝑥 ∖ {𝑧})))
 
Theoremkmlem12 9021* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.)
𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}       (∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ → (∀𝑧𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴)))))
 
Theoremkmlem13 9022* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 5-Apr-2004.)
𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}       (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥(¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
 
Theoremkmlem14 9023* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
(𝜑 ↔ (𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)))    &   (𝜓 ↔ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))    &   (𝜒 ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))       (∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ∃𝑦𝑧𝑣𝑢(𝑦𝑥𝜑))
 
Theoremkmlem15 9024* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
(𝜑 ↔ (𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)))    &   (𝜓 ↔ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))    &   (𝜒 ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))       ((¬ 𝑦𝑥𝜒) ↔ ∀𝑧𝑣𝑢𝑦𝑥𝜓))
 
Theoremkmlem16 9025* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
(𝜑 ↔ (𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)))    &   (𝜓 ↔ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))    &   (𝜒 ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))       ((∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥𝜒)) ↔ ∃𝑦𝑧𝑣𝑢((𝑦𝑥𝜑) ∨ (¬ 𝑦𝑥𝜓)))
 
Theoremdfackm 9026* Equivalence of the Axiom of Choice and Maes' AC ackm 9325. The proof consists of lemmas kmlem1 9010 through kmlem16 9025 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 8989 with biid 251) establishes the AC equivalence shown by Maes' writeup. The left-hand-side AC shown here was chosen because it is shorter to display. (Contributed by NM, 13-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ ∀𝑥𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))))
 
2.6.9  Cardinal number arithmetic
 
Syntaxccda 9027 Extend class definition to include cardinal number addition.
class +𝑐
 
Definitiondf-cda 9028* Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 9030 for its value and a description. (Contributed by NM, 24-Sep-2004.)
+𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})))
 
Theoremcdafn 9029 Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)
+𝑐 Fn (V × V)
 
Theoremcdaval 9030 Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while Cartesian product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 9411, carddom 9414, and cardsdom 9415. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
((𝐴𝑉𝐵𝑊) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
 
Theoremuncdadom 9031 Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ≼ (𝐴 +𝑐 𝐵))
 
Theoremcdaun 9032 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by NM, 5-Apr-2007.)
((𝐴𝑉𝐵𝑊 ∧ (𝐴𝐵) = ∅) → (𝐴 +𝑐 𝐵) ≈ (𝐴𝐵))
 
Theoremcdaen 9033 Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷))
 
Theoremcdaenun 9034 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴𝐵𝐶𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴 +𝑐 𝐶) ≈ (𝐵𝐷))
 
Theoremcda1en 9035 Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉 ∧ ¬ 𝐴𝐴) → (𝐴 +𝑐 1𝑜) ≈ suc 𝐴)
 
Theoremcda1dif 9036 Adding and subtracting one gives back the original set. Similar to pncan 10325 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
 
Theorempm110.643 9037 1+1=2 for cardinal number addition, derived from pm54.43 8864 as promised. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 8796), but after applying definitions, our theorem is equivalent. The comment for cdaval 9030 explains why we use instead of =. See pm110.643ALT 9038 for a shorter proof that doesn't use pm54.43 8864. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)
(1𝑜 +𝑐 1𝑜) ≈ 2𝑜
 
Theorempm110.643ALT 9038 Alternate proof of pm110.643 9037. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(1𝑜 +𝑐 1𝑜) ≈ 2𝑜
 
Theoremcda0en 9039 Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝑉 → (𝐴 +𝑐 ∅) ≈ 𝐴)
 
Theoremxp2cda 9040 Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝑉 → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴))
 
Theoremcdacomen 9041 Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)
 
Theoremcdaassen 9042 Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (𝐴 +𝑐 (𝐵 +𝑐 𝐶)))
 
Theoremxpcdaen 9043 Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) ≈ ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)))
 
Theoremmapcdaen 9044 Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))
 
Theorempwcdaen 9045 Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 +𝑐 𝐵) ≈ (𝒫 𝐴 × 𝒫 𝐵))
 
Theoremcdadom1 9046 Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝐵 → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
 
Theoremcdadom2 9047 Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝐵 → (𝐶 +𝑐 𝐴) ≼ (𝐶 +𝑐 𝐵))
 
Theoremcdadom3 9048 A set is dominated by its cardinal sum with another. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊) → 𝐴 ≼ (𝐴 +𝑐 𝐵))
 
Theoremcdaxpdom 9049 Cartesian product dominates disjoint union for sets with cardinality greater than 1. Similar to Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 18-May-2015.)
((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))
 
Theoremcdafi 9050 The cardinal sum of two finite sets is finite. (Contributed by NM, 22-Oct-2004.)
((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 +𝑐 𝐵) ≺ ω)
 
Theoremcdainflem 9051 Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.)
((𝐴𝐵) ≈ ω → (𝐴 ≈ ω ∨ 𝐵 ≈ ω))
 
Theoremcdainf 9052 A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(ω ≼ 𝐴 ↔ ω ≼ (𝐴 +𝑐 𝐴))
 
Theoreminfcda1 9053 An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)
(ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
 
Theorempwcda1 9054 The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.)
(𝐴𝑉 → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 (𝐴 +𝑐 1𝑜))
 
Theorempwcdaidm 9055 If the natural numbers inject into 𝐴, then 𝒫 𝐴 is idempotent under cardinal sum. (Contributed by Mario Carneiro, 15-May-2015.)
(ω ≼ 𝐴 → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴)
 
Theoremcdalepw 9056 If 𝐴 is idempotent under cardinal sum and 𝐵 is dominated by the power set of 𝐴, then so is the cardinal sum of 𝐴 and 𝐵. (Contributed by Mario Carneiro, 15-May-2015.)
(((𝐴 +𝑐 𝐴) ≈ 𝐴𝐵 ≼ 𝒫 𝐴) → (𝐴 +𝑐 𝐵) ≼ 𝒫 𝐴)
 
Theoremonacda 9057 The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ≈ (𝐴 +𝑐 𝐵))
 
Theoremcardacda 9058 The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
 
Theoremcdanum 9059 The cardinal sum of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ∈ dom card)
 
Theoremunnum 9060 The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ∈ dom card)
 
Theoremnnacda 9061 The cardinal and ordinal sums of finite ordinals are equal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (card‘(𝐴 +𝑐 𝐵)) = (𝐴 +𝑜 𝐵))
 
Theoremficardun 9062 The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (card‘(𝐴𝐵)) = ((card‘𝐴) +𝑜 (card‘𝐵)))
 
Theoremficardun2 9063 The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +𝑜 (card‘𝐵)))
 
Theorempwsdompw 9064* Lemma for domtriom 9303. This is the equinumerosity version of the algebraic identity Σ𝑘𝑛(2↑𝑘) = (2↑𝑛) − 1. (Contributed by Mario Carneiro, 7-Feb-2013.)
((𝑛 ∈ ω ∧ ∀𝑘 ∈ suc 𝑛(𝐵𝑘) ≈ 𝒫 𝑘) → 𝑘𝑛 (𝐵𝑘) ≺ (𝐵𝑛))
 
Theoremunctb 9065 The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴𝐵) ≼ ω)
 
Theoreminfcdaabs 9066 Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 +𝑐 𝐵) ≈ 𝐴)
 
Theoreminfunabs 9067 An infinite set is equinumerous to its union with a smaller one. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
 
Theoreminfcda 9068 The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≈ (𝐴𝐵))
 
Theoreminfdif 9069 The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
 
Theoreminfdif2 9070 Cardinality ordering for an infinite class difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ≼ 𝐵𝐴𝐵))
 
Theoreminfxpdom 9071 Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 × 𝐵) ≼ 𝐴)
 
Theoreminfxpabs 9072 Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
 
Theoreminfunsdom1 9073 The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 9074 assumes additionally that 𝐴 is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)
(((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
 
Theoreminfunsdom 9074 The union of two sets that are strictly dominated by the infinite set 𝑋 is also strictly dominated by 𝑋. (Contributed by Mario Carneiro, 3-May-2015.)
(((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
 
Theoreminfxp 9075 Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
 
Theorempwcdadom 9076 A property of dominance over a powerset, and a main lemma for gchac 9541. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
(𝒫 (𝐴 +𝑐 𝐴) ≼ (𝐴 +𝑐 𝐵) → 𝒫 𝐴𝐵)
 
Theoreminfpss 9077* Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 9173. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
(ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
 
Theoreminfmap2 9078* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 9436 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
 
2.6.10  The Ackermann bijection
 
Theoremackbij2lem1 9079 Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))
 
Theoremackbij1lem1 9080 Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐴𝐵 → (𝐵 ∩ suc 𝐴) = (𝐵𝐴))
 
Theoremackbij1lem2 9081 Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝐴𝐵 → (𝐵 ∩ suc 𝐴) = ({𝐴} ∪ (𝐵𝐴)))
 
Theoremackbij1lem3 9082 Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))
 
Theoremackbij1lem4 9083 Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 19-Nov-2014.)
(𝐴 ∈ ω → {𝐴} ∈ (𝒫 ω ∩ Fin))
 
Theoremackbij1lem5 9084 Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 19-Nov-2014.)
(𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴)))
 
Theoremackbij1lem6 9085 Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 18-Nov-2014.)
((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴𝐵) ∈ (𝒫 ω ∩ Fin))
 
Theoremackbij1lem7 9086* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 21-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
 
Theoremackbij1lem8 9087* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 19-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
 
Theoremackbij1lem9 9088* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 19-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (𝐹‘(𝐴𝐵)) = ((𝐹𝐴) +𝑜 (𝐹𝐵)))
 
Theoremackbij1lem10 9089* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       𝐹:(𝒫 ω ∩ Fin)⟶ω
 
Theoremackbij1lem11 9090* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵𝐴) → 𝐵 ∈ (𝒫 ω ∩ Fin))
 
Theoremackbij1lem12 9091* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))
 
Theoremackbij1lem13 9092* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐹‘∅) = ∅
 
Theoremackbij1lem14 9093* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
 
Theoremackbij1lem15 9094* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝑐 ∈ ω ∧ 𝑐𝐴 ∧ ¬ 𝑐𝐵)) → ¬ (𝐹‘(𝐴 ∩ suc 𝑐)) = (𝐹‘(𝐵 ∩ suc 𝑐)))
 
Theoremackbij1lem16 9095* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
 
Theoremackbij1lem17 9096* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       𝐹:(𝒫 ω ∩ Fin)–1-1→ω
 
Theoremackbij1lem18 9097* Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝐴))
 
Theoremackbij1 9098* The Ackermann bijection, part 1: each natural number can be uniquely coded in binary as a finite set of natural numbers and conversely. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω
 
Theoremackbij1b 9099* The Ackermann bijection, part 1b: the bijection from ackbij1 9098 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴))
 
Theoremackbij2lem2 9100* Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))    &   𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))       (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴):(𝑅1𝐴)–1-1-onto→(card‘(𝑅1𝐴)))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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