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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cardalephex 9501* | Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse. (Contributed by NM, 5-Nov-2003.) |
⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))) | ||
Theorem | infenaleph 9502* | An infinite numerable set is equinumerous to an infinite initial ordinal. (Contributed by Jeff Hankins, 23-Oct-2009.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴) | ||
Theorem | isinfcard 9503 | Two ways to express the property of being a transfinite cardinal. (Contributed by NM, 9-Nov-2003.) |
⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ) | ||
Theorem | iscard3 9504 | Two ways to express the property of being a cardinal number. (Contributed by NM, 9-Nov-2003.) |
⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ (ω ∪ ran ℵ)) | ||
Theorem | cardnum 9505 | Two ways to express the class of all cardinal numbers, which consists of the finite ordinals in ω plus the transfinite alephs. (Contributed by NM, 10-Sep-2004.) |
⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} = (ω ∪ ran ℵ) | ||
Theorem | alephinit 9506* | An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥))) | ||
Theorem | carduniima 9507 | The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.) |
⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))) | ||
Theorem | cardinfima 9508* | If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.) |
⊢ (𝐴 ∈ 𝐵 → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ ran ℵ)) | ||
Theorem | alephiso 9509 | Aleph is an order isomorphism of the class of ordinal numbers onto the class of infinite cardinals. Definition 10.27 of [TakeutiZaring] p. 90. (Contributed by NM, 3-Aug-2004.) |
⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}) | ||
Theorem | alephprc 9510 | The class of all transfinite cardinal numbers (the range of the aleph function) is a proper class. Proposition 10.26 of [TakeutiZaring] p. 90. (Contributed by NM, 11-Nov-2003.) |
⊢ ¬ ran ℵ ∈ V | ||
Theorem | alephsson 9511 | The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.) |
⊢ ran ℵ ⊆ On | ||
Theorem | unialeph 9512 | The union of the class of transfinite cardinals (the range of the aleph function) is the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.) |
⊢ ∪ ran ℵ = On | ||
Theorem | alephsmo 9513 | The aleph function is strictly monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) |
⊢ Smo ℵ | ||
Theorem | alephf1ALT 9514 | Alternate proof of alephf1 9496. (Contributed by Mario Carneiro, 15-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℵ:On–1-1→On | ||
Theorem | alephfplem1 9515 | Lemma for alephfp 9519. (Contributed by NM, 6-Nov-2004.) |
⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) ⇒ ⊢ (𝐻‘∅) ∈ ran ℵ | ||
Theorem | alephfplem2 9516* | Lemma for alephfp 9519. (Contributed by NM, 6-Nov-2004.) |
⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) ⇒ ⊢ (𝑤 ∈ ω → (𝐻‘suc 𝑤) = (ℵ‘(𝐻‘𝑤))) | ||
Theorem | alephfplem3 9517* | Lemma for alephfp 9519. (Contributed by NM, 6-Nov-2004.) |
⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) ⇒ ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ) | ||
Theorem | alephfplem4 9518 | Lemma for alephfp 9519. (Contributed by NM, 5-Nov-2004.) |
⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) ⇒ ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ | ||
Theorem | alephfp 9519 | The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 9520 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.) |
⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) ⇒ ⊢ (ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) | ||
Theorem | alephfp2 9520 | The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 9519 for an actual example of a fixed point. Compare the inequality alephle 9499 that holds in general. Note that if 𝑥 is a fixed point, then ℵ‘ℵ‘ℵ‘... ℵ‘𝑥 = 𝑥. (Contributed by NM, 6-Nov-2004.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝑥 | ||
Theorem | alephval3 9521* | An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 . (Contributed by NM, 16-Nov-2003.) |
⊢ (𝐴 ∈ On → (ℵ‘𝐴) = ∩ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))}) | ||
Theorem | alephsucpw2 9522 | The power set of an aleph is not strictly dominated by the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 10087 or gchaleph2 10083.) The transposed form alephsucpw 9981 cannot be proven without the AC, and is in fact equivalent to it. (Contributed by Mario Carneiro, 2-Feb-2013.) |
⊢ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) | ||
Theorem | mappwen 9523 | Power rule for cardinal arithmetic. Theorem 11.21 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 27-Apr-2015.) |
⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (𝐴 ↑m 𝐵) ≈ 𝒫 𝐵) | ||
Theorem | finnisoeu 9524* | A finite totally ordered set has a unique order isomorphism to a finite ordinal. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 26-Jun-2015.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴)) | ||
Theorem | iunfictbso 9525 | Countability of a countable union of finite sets with a strict (not globally well) order fulfilling the choice role. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ ((𝐴 ≼ ω ∧ 𝐴 ⊆ Fin ∧ 𝐵 Or ∪ 𝐴) → ∪ 𝐴 ≼ ω) | ||
Syntax | wac 9526 | Wff for an abbreviation of the axiom of choice. |
wff CHOICE | ||
Definition | df-ac 9527* |
The expression CHOICE will be used as a
readable shorthand for any
form of the axiom of choice; all concrete forms are long, cryptic, have
dummy variables, or all three, making it useful to have a short name.
Similar to the Axiom of Choice (first form) of [Enderton] p. 49.
There is a slight problem with taking the exact form of ax-ac 9870 as our definition, because the equivalence to more standard forms (dfac2 9542) requires the Axiom of Regularity, which we often try to avoid. Thus, we take the first of the "textbook forms" as the definition and derive the form of ax-ac 9870 itself as dfac0 9544. (Contributed by Mario Carneiro, 22-Feb-2015.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)) | ||
Theorem | aceq1 9528* | Equivalence of two versions of the Axiom of Choice ax-ac 9870. The proof uses neither AC nor the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥))) | ||
Theorem | aceq0 9529* | Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac 9870. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))) | ||
Theorem | aceq2 9530* | Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) | ||
Theorem | aceq3lem 9531* | Lemma for dfac3 9532. (Contributed by NM, 2-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝐹 = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})) ⇒ ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦)) | ||
Theorem | dfac3 9532* | Equivalence of two versions of the Axiom of Choice. The left-hand side is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is the Axiom of Choice of [TakeutiZaring] p. 83. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Stefan O'Rear, 22-Feb-2015.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) | ||
Theorem | dfac4 9533* | Equivalence of two versions of the Axiom of Choice. The right-hand side is Axiom AC of [BellMachover] p. 488. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))) | ||
Theorem | dfac5lem1 9534* | Lemma for dfac5 9539. (Contributed by NM, 12-Apr-2004.) |
⊢ (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔 ∈ 𝑤 ∧ 〈𝑤, 𝑔〉 ∈ 𝑦)) | ||
Theorem | dfac5lem2 9535* | Lemma for dfac5 9539. (Contributed by NM, 12-Apr-2004.) |
⊢ 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} ⇒ ⊢ (〈𝑤, 𝑔〉 ∈ ∪ 𝐴 ↔ (𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤)) | ||
Theorem | dfac5lem3 9536* | Lemma for dfac5 9539. (Contributed by NM, 12-Apr-2004.) |
⊢ 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} ⇒ ⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ)) | ||
Theorem | dfac5lem4 9537* | Lemma for dfac5 9539. (Contributed by NM, 11-Apr-2004.) |
⊢ 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} & ⊢ 𝐵 = (∪ 𝐴 ∩ 𝑦) & ⊢ (𝜑 ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ⇒ ⊢ (𝜑 → ∃𝑦∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) | ||
Theorem | dfac5lem5 9538* | Lemma for dfac5 9539. (Contributed by NM, 12-Apr-2004.) |
⊢ 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} & ⊢ 𝐵 = (∪ 𝐴 ∩ 𝑦) & ⊢ (𝜑 ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ⇒ ⊢ (𝜑 → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) | ||
Theorem | dfac5 9539* | Equivalence of two versions of the Axiom of Choice. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ (CHOICE ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) | ||
Theorem | dfac2a 9540* | Our Axiom of Choice (in the form of ac3 9873) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2b 9541 for the converse (which does use the Axiom of Regularity). (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → CHOICE) | ||
Theorem | dfac2b 9541* | Axiom of Choice (first form) of [Enderton] p. 49 implies our Axiom of Choice (in the form of ac3 9873). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elneq 9046 and preleq 9063 that are referenced in the proof each make use of Regularity for their derivations. (The reverse implication can be derived without using Regularity; see dfac2a 9540.) (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 16-Jun-2022.) |
⊢ (CHOICE → ∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) | ||
Theorem | dfac2 9542* | Axiom of Choice (first form) of [Enderton] p. 49 corresponds to our Axiom of Choice (in the form of ac3 9873). The proof does not make use of AC, but the Axiom of Regularity is used (by applying dfac2b 9541). (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 16-Jun-2022.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))) | ||
Theorem | dfac7 9543* | Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 9872). The proof does not depend on AC but does depend on the Axiom of Regularity. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) | ||
Theorem | dfac0 9544* | Equivalence of two versions of the Axiom of Choice. The proof uses the Axiom of Regularity. The right-hand side is our original ax-ac 9870. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))) | ||
Theorem | dfac1 9545* | Equivalence of two versions of the Axiom of Choice ax-ac 9870. The proof uses the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥))) | ||
Theorem | dfac8 9546* | A proof of the equivalency of the well-ordering theorem weth 9906 and the axiom of choice ac7 9884. (Contributed by Mario Carneiro, 5-Jan-2013.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑟 𝑟 We 𝑥) | ||
Theorem | dfac9 9547* | Equivalence of the axiom of choice with a statement related to ac9 9894; definition AC3 of [Schechter] p. 139. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (CHOICE ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅)) | ||
Theorem | dfac10 9548 | Axiom of Choice equivalent: the cardinality function measures every set. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ (CHOICE ↔ dom card = V) | ||
Theorem | dfac10c 9549* | Axiom of Choice equivalent: every set is equinumerous to an ordinal. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑦 ∈ On 𝑦 ≈ 𝑥) | ||
Theorem | dfac10b 9550 | Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 9527). (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ (CHOICE ↔ ( ≈ “ On) = V) | ||
Theorem | acacni 9551 | A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → AC 𝐴 = V) | ||
Theorem | dfacacn 9552 | A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (CHOICE ↔ ∀𝑥AC 𝑥 = V) | ||
Theorem | dfac13 9553 | The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (CHOICE ↔ ∀𝑥 𝑥 ∈ AC 𝑥) | ||
Theorem | dfac12lem1 9554* | Lemma for dfac12 9560. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–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 → 𝐶 ∈ On) & ⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)) ⇒ ⊢ (𝜑 → (𝐺‘𝐶) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))))) | ||
Theorem | dfac12lem2 9555* | Lemma for dfac12 9560. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–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 → 𝐶 ∈ On) & ⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On) ⇒ ⊢ (𝜑 → (𝐺‘𝐶):(𝑅1‘𝐶)–1-1→On) | ||
Theorem | dfac12lem3 9556* | Lemma for dfac12 9560. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–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 → (𝑅1‘𝐴) ∈ dom card) | ||
Theorem | dfac12r 9557 | The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 9560 does not assume the Axiom of Regularity. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∪ (𝑅1 “ On) ⊆ dom card) | ||
Theorem | dfac12k 9558* | Equivalence of dfac12 9560 and dfac12a 9559, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.) |
⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card) | ||
Theorem | dfac12a 9559 | 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) | ||
Theorem | dfac12 9560 | 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) | ||
Theorem | kmlem1 9561* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, 1 => 2. (Contributed by NM, 5-Apr-2004.) |
⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 𝜑) → ∃𝑦∀𝑧 ∈ 𝑥 𝜓) → ∀𝑥(∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 𝜑 → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → 𝜓))) | ||
Theorem | kmlem2 9562* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.) |
⊢ (∃𝑦∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)) ↔ ∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)))) | ||
Theorem | kmlem3 9563* | 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})) ≠ ∅ ↔ ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤))) | ||
Theorem | kmlem4 9564* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.) |
⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅) | ||
Theorem | kmlem5 9565* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.) |
⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅) | ||
Theorem | kmlem6 9566* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.) |
⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)) → ∀𝑧 ∈ 𝑥 ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴)) | ||
Theorem | kmlem7 9567* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.) |
⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ¬ ∃𝑧 ∈ 𝑥 ∀𝑣 ∈ 𝑧 ∃𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 ∧ 𝑣 ∈ (𝑧 ∩ 𝑤))) | ||
Theorem | kmlem8 9568* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 4-Apr-2004.) |
⊢ ((¬ ∃𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑧 𝜓 → ∃𝑦∀𝑧 ∈ 𝑢 (𝑧 ≠ ∅ → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦))) ↔ (∃𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑧 𝜓 ∨ ∃𝑦(¬ 𝑦 ∈ 𝑢 ∧ ∀𝑧 ∈ 𝑢 ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)))) | ||
Theorem | kmlem9 9569* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.) |
⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ⇒ ⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) | ||
Theorem | kmlem10 9570* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.) |
⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ⇒ ⊢ (∀ℎ(∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ ℎ 𝜑) → ∃𝑦∀𝑧 ∈ 𝐴 𝜑) | ||
Theorem | kmlem11 9571* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.) |
⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ⇒ ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ ∪ 𝐴) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) | ||
Theorem | kmlem12 9572* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.) |
⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ⇒ ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ → (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴))))) | ||
Theorem | kmlem13 9573* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 5-Apr-2004.) |
⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ⇒ ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥(¬ ∃𝑧 ∈ 𝑥 ∀𝑣 ∈ 𝑧 ∃𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 ∧ 𝑣 ∈ (𝑧 ∩ 𝑤)) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))) | ||
Theorem | kmlem14 9574* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.) |
⊢ (𝜑 ↔ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ 𝑦 ≠ 𝑣) ∧ 𝑧 ∈ 𝑣))) & ⊢ (𝜓 ↔ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))) & ⊢ (𝜒 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ⇒ ⊢ (∃𝑧 ∈ 𝑥 ∀𝑣 ∈ 𝑧 ∃𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 ∧ 𝑣 ∈ (𝑧 ∩ 𝑤)) ↔ ∃𝑦∀𝑧∃𝑣∀𝑢(𝑦 ∈ 𝑥 ∧ 𝜑)) | ||
Theorem | kmlem15 9575* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.) |
⊢ (𝜑 ↔ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ 𝑦 ≠ 𝑣) ∧ 𝑧 ∈ 𝑣))) & ⊢ (𝜓 ↔ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))) & ⊢ (𝜒 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ⇒ ⊢ ((¬ 𝑦 ∈ 𝑥 ∧ 𝜒) ↔ ∀𝑧∃𝑣∀𝑢(¬ 𝑦 ∈ 𝑥 ∧ 𝜓)) | ||
Theorem | kmlem16 9576* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4. (Contributed by NM, 4-Apr-2004.) |
⊢ (𝜑 ↔ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ 𝑦 ≠ 𝑣) ∧ 𝑧 ∈ 𝑣))) & ⊢ (𝜓 ↔ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))) & ⊢ (𝜒 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ⇒ ⊢ ((∃𝑧 ∈ 𝑥 ∀𝑣 ∈ 𝑧 ∃𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 ∧ 𝑣 ∈ (𝑧 ∩ 𝑤)) ∨ ∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ 𝜒)) ↔ ∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ 𝜑) ∨ (¬ 𝑦 ∈ 𝑥 ∧ 𝜓))) | ||
Theorem | dfackm 9577* | Equivalence of the Axiom of Choice and Maes' AC ackm 9876. The proof consists of lemmas kmlem1 9561 through kmlem16 9576 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 9539 with biid 264) 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 ↔ ∀𝑥∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))) | ||
For cardinal arithmetic, we follow [Mendelson] p. 258. Rather than defining operations restricted to cardinal numbers, we use disjoint union df-dju 9314 (⊔) for cardinal addition, Cartesian product df-xp 5525 (×) for cardinal multiplication, and set exponentiation df-map 8391 (↑m) for cardinal 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 9962, carddom 9965, and cardsdom 9966. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available. | ||
Theorem | undjudom 9578 | Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.) (Revised by Jim Kingdon, 15-Aug-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) | ||
Theorem | endjudisj 9579 | Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by NM, 5-Apr-2007.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵)) | ||
Theorem | djuen 9580 | Disjoint unions of equinumerous sets are equinumerous. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷)) | ||
Theorem | djuenun 9581 | Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ∪ 𝐷)) | ||
Theorem | dju1en 9582 | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 ⊔ 1o) ≈ suc 𝐴) | ||
Theorem | dju1dif 9583 | Adding and subtracting one gives back the original cardinality. Similar to pncan 10881 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by Jim Kingdon, 20-Aug-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {𝐵}) ≈ 𝐴) | ||
Theorem | dju1p1e2 9584 | 1+1=2 for cardinal number addition, derived from pm54.43 9414 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 9308), but after applying definitions, our theorem is equivalent. Because we use a disjoint union for cardinal addition (as explained in the comment at the top of this section), we use ≈ instead of =. See dju1p1e2ALT 9585 for a shorter proof that doesn't use pm54.43 9414. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.) |
⊢ (1o ⊔ 1o) ≈ 2o | ||
Theorem | dju1p1e2ALT 9585 | Alternate proof of dju1p1e2 9584. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (1o ⊔ 1o) ≈ 2o | ||
Theorem | dju0en 9586 | 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ 𝐴) | ||
Theorem | xp2dju 9587 | 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.) |
⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) | ||
Theorem | djucomen 9588 | 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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴)) | ||
Theorem | djuassen 9589 | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵 ⊔ 𝐶))) | ||
Theorem | xpdjuen 9590 | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 ⊔ 𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶))) | ||
Theorem | mapdjuen 9591 | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ↑m (𝐵 ⊔ 𝐶)) ≈ ((𝐴 ↑m 𝐵) × (𝐴 ↑m 𝐶))) | ||
Theorem | pwdjuen 9592 | Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 ⊔ 𝐵) ≈ (𝒫 𝐴 × 𝒫 𝐵)) | ||
Theorem | djudom1 9593 | 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.) (Revised by Jim Kingdon, 1-Sep-2023.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝐴 ⊔ 𝐶) ≼ (𝐵 ⊔ 𝐶)) | ||
Theorem | djudom2 9594 | 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.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝐶 ⊔ 𝐴) ≼ (𝐶 ⊔ 𝐵)) | ||
Theorem | djudoml 9595 | A set is dominated by its disjoint union with another. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ≼ (𝐴 ⊔ 𝐵)) | ||
Theorem | djuxpdom 9596 | 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.) |
⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (𝐴 × 𝐵)) | ||
Theorem | djufi 9597 | The disjoint union of two finite sets is finite. (Contributed by NM, 22-Oct-2004.) |
⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ⊔ 𝐵) ≺ ω) | ||
Theorem | cdainflem 9598 | Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.) |
⊢ ((𝐴 ∪ 𝐵) ≈ ω → (𝐴 ≈ ω ∨ 𝐵 ≈ ω)) | ||
Theorem | djuinf 9599 | 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 ⊔ 𝐴)) | ||
Theorem | infdju1 9600 | An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 1o) ≈ 𝐴) |
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