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Type | Label | Description |
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Statement | ||
Theorem | rankr1 9601 | A relationship between the rank function and the cumulative hierarchy of sets function 𝑅1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by NM, 6-Oct-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))) | ||
Theorem | ssrankr1 9602 | A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets 𝑅1. Proposition 9.15(3) of [TakeutiZaring] p. 79. (Contributed by NM, 8-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1‘𝐵))) | ||
Theorem | rankr1a 9603 | A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 9602 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 9631 for the subset version. (Contributed by Raph Levien, 29-May-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) | ||
Theorem | r1val2 9604* | The value of the cumulative hierarchy of sets function expressed in terms of rank. Definition 15.19 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) |
⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = {𝑥 ∣ (rank‘𝑥) ∈ 𝐴}) | ||
Theorem | r1val3 9605* | The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥}) | ||
Theorem | rankel 9606 | The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)) | ||
Theorem | rankval3 9607* | The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} | ||
Theorem | bndrank 9608* | Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.) |
⊢ (∃𝑥 ∈ On ∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V) | ||
Theorem | unbndrank 9609* | The elements of a proper class have unbounded rank. Exercise 2 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.) |
⊢ (¬ 𝐴 ∈ V → ∀𝑥 ∈ On ∃𝑦 ∈ 𝐴 𝑥 ∈ (rank‘𝑦)) | ||
Theorem | rankpw 9610 | The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 22-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘𝒫 𝐴) = suc (rank‘𝐴) | ||
Theorem | ranklim 9611 | The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.) |
⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) | ||
Theorem | r1pw 9612 | A stronger property of 𝑅1 than rankpw 9610. The latter merely proves that 𝑅1 of the successor is a power set, but here we prove that if 𝐴 is in the cumulative hierarchy, then 𝒫 𝐴 is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))) | ||
Theorem | r1pwALT 9613 | Alternate shorter proof of r1pw 9612 based on the additional axioms ax-reg 9360 and ax-inf2 9408. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))) | ||
Theorem | r1pwcl 9614 | The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2014.) |
⊢ (Lim 𝐵 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵))) | ||
Theorem | rankssb 9615 | The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵))) | ||
Theorem | rankss 9616 | The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)) | ||
Theorem | rankunb 9617 | The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))) | ||
Theorem | rankprb 9618 | The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))) | ||
Theorem | rankopb 9619 | The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))) | ||
Theorem | rankuni2b 9620* | The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 8-Jun-2013.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)) | ||
Theorem | ranksn 9621 | The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘{𝐴}) = suc (rank‘𝐴) | ||
Theorem | rankuni2 9622* | The rank of a union. Part of Theorem 15.17(iv) of [Monk1] p. 112. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) | ||
Theorem | rankun 9623 | The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by NM, 26-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)) | ||
Theorem | rankpr 9624 | The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) | ||
Theorem | rankop 9625 | The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 13-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)) | ||
Theorem | r1rankid 9626 | Any set is a subset of the hierarchy of its rank. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | ||
Theorem | rankeq0b 9627 | A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅)) | ||
Theorem | rankeq0 9628 | A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 = ∅ ↔ (rank‘𝐴) = ∅) | ||
Theorem | rankr1id 9629 | The rank of the hierarchy of an ordinal number is itself. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝐴)) = 𝐴) | ||
Theorem | rankuni 9630 | The rank of a union. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴) | ||
Theorem | rankr1b 9631 | A relationship between rank and 𝑅1. See rankr1a 9603 for the membership version. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ On → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) | ||
Theorem | ranksuc 9632 | The rank of a successor. (Contributed by NM, 18-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘suc 𝐴) = suc (rank‘𝐴) | ||
Theorem | rankuniss 9633 | Upper bound of the rank of a union. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 30-Nov-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘∪ 𝐴) ⊆ (rank‘𝐴) | ||
Theorem | rankval4 9634* | The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. (Contributed by NM, 12-Oct-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) | ||
Theorem | rankbnd 9635* | The rank of a set is bounded by a bound for the successor of its members. (Contributed by NM, 18-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ 𝐵) | ||
Theorem | rankbnd2 9636* | The rank of a set is bounded by the successor of a bound for its members. (Contributed by NM, 15-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ On → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵)) | ||
Theorem | rankc1 9637* | A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ (rank‘𝐴) = (rank‘∪ 𝐴)) | ||
Theorem | rankc2 9638* | A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴)) | ||
Theorem | rankelun 9639 | Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷))) | ||
Theorem | rankelpr 9640 | Rank membership is inherited by unordered pairs. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷})) | ||
Theorem | rankelop 9641 | Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘〈𝐴, 𝐵〉) ∈ (rank‘〈𝐶, 𝐷〉)) | ||
Theorem | rankxpl 9642 | A lower bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))) | ||
Theorem | rankxpu 9643 | An upper bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) | ||
Theorem | rankfu 9644 | An upper bound on the rank of a function. (Contributed by Gérard Lang, 5-Aug-2018.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹:𝐴⟶𝐵 → (rank‘𝐹) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))) | ||
Theorem | rankmapu 9645 | An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (rank‘(𝐴 ↑m 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) | ||
Theorem | rankxplim 9646 | The rank of a Cartesian product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 9649 for the successor case. (Contributed by NM, 19-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵))) | ||
Theorem | rankxplim2 9647 | If the rank of a Cartesian product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵))) | ||
Theorem | rankxplim3 9648 | The rank of a Cartesian product is a limit ordinal iff its union is. (Contributed by NM, 19-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵))) | ||
Theorem | rankxpsuc 9649 | The rank of a Cartesian product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 9646 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵))) | ||
Theorem | tcwf 9650 | The transitive closure function is well-founded if its argument is. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ∈ ∪ (𝑅1 “ On)) | ||
Theorem | tcrank 9651 | This theorem expresses two different facts from the two subset implications in this equality. In the forward direction, it says that the transitive closure has members of every rank below 𝐴. Stated another way, to construct a set at a given rank, you have to climb the entire hierarchy of ordinals below (rank‘𝐴), constructing at least one set at each level in order to move up the ranks. In the reverse direction, it says that every member of (TC‘𝐴) has a rank below the rank of 𝐴, since intuitively it contains only the members of 𝐴 and the members of those and so on, but nothing "bigger" than 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴))) | ||
Theorem | scottex 9652* | Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set. (Contributed by NM, 13-Oct-2003.) |
⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V | ||
Theorem | scott0 9653* | Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. 𝐴 is empty). (Contributed by NM, 15-Oct-2003.) |
⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅) | ||
Theorem | scottexs 9654* | Theorem scheme version of scottex 9652. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is a set. (Contributed by NM, 13-Oct-2003.) |
⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V | ||
Theorem | scott0s 9655* | Theorem scheme version of scott0 9653. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is not empty iff there is an 𝑥 such that 𝜑(𝑥) holds. (Contributed by NM, 13-Oct-2003.) |
⊢ (∃𝑥𝜑 ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅) | ||
Theorem | cplem1 9656* | Lemma for the Collection Principle cp 9658. (Contributed by NM, 17-Oct-2003.) |
⊢ 𝐶 = {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} & ⊢ 𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶 ⇒ ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅) | ||
Theorem | cplem2 9657* | Lemma for the Collection Principle cp 9658. (Contributed by NM, 17-Oct-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅) | ||
Theorem | cp 9658* | Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 9652 that collapses a proper class into a set of minimum rank. The wff 𝜑 can be thought of as 𝜑(𝑥, 𝑦). Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.) |
⊢ ∃𝑤∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) | ||
Theorem | bnd 9659* | A very strong generalization of the Axiom of Replacement (compare zfrep6 7806), derived from the Collection Principle cp 9658. Its strength lies in the rather profound fact that 𝜑(𝑥, 𝑦) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. (Contributed by NM, 17-Oct-2004.) |
⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑) | ||
Theorem | bnd2 9660* | A variant of the Boundedness Axiom bnd 9659 that picks a subset 𝑧 out of a possibly proper class 𝐵 in which a property is true. (Contributed by NM, 4-Feb-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑)) | ||
Theorem | kardex 9661* | The collection of all sets equinumerous to a set 𝐴 and having the least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.) |
⊢ {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V | ||
Theorem | karden 9662* | If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 10316). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 9661 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥 ∣ 𝑥 ≈ 𝐴}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.) |
⊢ 𝐴 ∈ V & ⊢ 𝐶 = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ⇒ ⊢ (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵) | ||
Theorem | htalem 9663* | Lemma for defining an emulation of Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem is equivalent to Hilbert's "transfinite axiom", described on that page, with the additional 𝑅 We 𝐴 antecedent. The element 𝐵 is the epsilon that the theorem emulates. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴) | ||
Theorem | hta 9664* |
A ZFC emulation of Hilbert's transfinite axiom. The set 𝐵 has the
properties of Hilbert's epsilon, except that it also depends on a
well-ordering 𝑅. This theorem arose from
discussions with Raph
Levien on 5-Mar-2004 about translating the HOL proof language, which
uses Hilbert's epsilon. See
https://us.metamath.org/downloads/choice.txt
(copy of obsolete link
http://ghilbert.org/choice.txt) and
https://us.metamath.org/downloads/megillaward2005he.pdf.
Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires 𝑅 We 𝐴 as an antecedent. Class 𝐴 collects the sets of the least rank for which 𝜑(𝑥) is true. Class 𝐵, which emulates Hilbert's epsilon, is the minimum element in a well-ordering 𝑅 on 𝐴. If a well-ordering 𝑅 on 𝐴 can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace 𝑅 with a dummy setvar variable, say 𝑤, and attach 𝑤 We 𝐴 as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, 𝐵 (which will have 𝑤 as a free variable) will no longer be present, and we can eliminate 𝑤 We 𝐴 by applying exlimiv 1934 and weth 10260, using scottexs 9654 to establish the existence of 𝐴. For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9663. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐴 = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} & ⊢ 𝐵 = (℩𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧) ⇒ ⊢ (𝑅 We 𝐴 → (𝜑 → [𝐵 / 𝑥]𝜑)) | ||
Syntax | cdju 9665 | Extend class notation to include disjoint union of two classes. |
class (𝐴 ⊔ 𝐵) | ||
Syntax | cinl 9666 | Extend class notation to include left injection of a disjoint union. |
class inl | ||
Syntax | cinr 9667 | Extend class notation to include right injection of a disjoint union. |
class inr | ||
Definition | df-dju 9668 | Disjoint union of two classes. This is a way of creating a set which contains elements corresponding to each element of 𝐴 or 𝐵, tagging each one with whether it came from 𝐴 or 𝐵. (Contributed by Jim Kingdon, 20-Jun-2022.) |
⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | ||
Definition | df-inl 9669 | Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | ||
Definition | df-inr 9670 | Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | ||
Theorem | djueq12 9671 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷)) | ||
Theorem | djueq1 9672 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) | ||
Theorem | djueq2 9673 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵)) | ||
Theorem | nfdju 9674 | Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵) | ||
Theorem | djuex 9675 | The disjoint union of sets is a set. For a shorter proof using djuss 9687 see djuexALT 9689. (Contributed by AV, 28-Jun-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) | ||
Theorem | djuexb 9676 | The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V) | ||
Theorem | djulcl 9677 | Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) | ||
Theorem | djurcl 9678 | Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵)) | ||
Theorem | djulf1o 9679 | The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
⊢ inl:V–1-1-onto→({∅} × V) | ||
Theorem | djurf1o 9680 | The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
⊢ inr:V–1-1-onto→({1o} × V) | ||
Theorem | inlresf 9681 | The left injection restricted to the left class of a disjoint union is a function from the left class into the disjoint union. (Contributed by AV, 27-Jun-2022.) |
⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) | ||
Theorem | inlresf1 9682 | The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.) |
⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) | ||
Theorem | inrresf 9683 | The right injection restricted to the right class of a disjoint union is a function from the right class into the disjoint union. (Contributed by AV, 27-Jun-2022.) |
⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) | ||
Theorem | inrresf1 9684 | The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.) |
⊢ (inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵) | ||
Theorem | djuin 9685 | The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) |
⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅ | ||
Theorem | djur 9686* | A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))) | ||
Theorem | djuss 9687 | A disjoint union is a subclass of a Cartesian product. (Contributed by AV, 25-Jun-2022.) |
⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) | ||
Theorem | djuunxp 9688 | The union of a disjoint union and its inversion is the Cartesian product of an unordered pair and the union of the left and right classes of the disjoint unions. (Proposed by GL, 4-Jul-2022.) (Contributed by AV, 4-Jul-2022.) |
⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) = ({∅, 1o} × (𝐴 ∪ 𝐵)) | ||
Theorem | djuexALT 9689 | Alternate proof of djuex 9675, which is shorter, but based indirectly on the definitions of inl and inr. (Proposed by BJ, 28-Jun-2022.) (Contributed by AV, 28-Jun-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) | ||
Theorem | eldju1st 9690 | The first component of an element of a disjoint union is either ∅ or 1o. (Contributed by AV, 26-Jun-2022.) |
⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o)) | ||
Theorem | eldju2ndl 9691 | The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.) |
⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) = ∅) → (2nd ‘𝑋) ∈ 𝐴) | ||
Theorem | eldju2ndr 9692 | The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.) |
⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵) | ||
Theorem | djuun 9693 | The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) |
⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵) | ||
Theorem | 1stinl 9694 | The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.) |
⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅) | ||
Theorem | 2ndinl 9695 | The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.) |
⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋) | ||
Theorem | 1stinr 9696 | The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.) |
⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o) | ||
Theorem | 2ndinr 9697 | The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.) |
⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋) | ||
Theorem | updjudhf 9698* | The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) & ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶) | ||
Theorem | updjudhcoinlf 9699* | The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) & ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ⇒ ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹) | ||
Theorem | updjudhcoinrg 9700* | The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) & ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ⇒ ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺) |
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