![]() |
Metamath
Proof Explorer Theorem List (p. 97 of 491) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30946) |
![]() (30947-32469) |
![]() (32470-49035) |
Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cwdom 9601 | Class symbol for the weak dominance relation. |
class ≼* | ||
Definition | df-wdom 9602* | A set is weakly dominated by a "larger" set if the "larger" set can be mapped onto the "smaller" set or the smaller set is empty, or equivalently, if the smaller set can be placed into bijection with some partition of the larger set. Dominance (df-dom 8985) implies weak dominance (over ZF). The principle asserting the converse is known as the partition principle and is independent of ZF. Theorem fodom 10560 proves that the axiom of choice implies the partition principle (over ZF). It is not known whether the partition principle is equivalent to the axiom of choice (over ZF), although it is know to imply dependent choice. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ ≼* = {〈𝑥, 𝑦〉 ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)} | ||
Theorem | relwdom 9603 | Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ Rel ≼* | ||
Theorem | brwdom 9604* | Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | ||
Theorem | brwdomi 9605* | Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.) |
⊢ (𝑋 ≼* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)) | ||
Theorem | brwdomn0 9606* | Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) | ||
Theorem | 0wdom 9607 | Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋) | ||
Theorem | fowdom 9608 | An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌) | ||
Theorem | wdomref 9609 | Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑋 ∈ 𝑉 → 𝑋 ≼* 𝑋) | ||
Theorem | brwdom2 9610* | Alternate characterization of the weak dominance predicate which does not require special treatment of the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋)) | ||
Theorem | domwdom 9611 | Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) | ||
Theorem | wdomtr 9612 | Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ ((𝑋 ≼* 𝑌 ∧ 𝑌 ≼* 𝑍) → 𝑋 ≼* 𝑍) | ||
Theorem | wdomen1 9613 | Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼* 𝐶 ↔ 𝐵 ≼* 𝐶)) | ||
Theorem | wdomen2 9614 | Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼* 𝐴 ↔ 𝐶 ≼* 𝐵)) | ||
Theorem | wdompwdom 9615 | Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌) | ||
Theorem | canthwdom 9616 | Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9168, equivalent to canth 7384). (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ¬ 𝒫 𝐴 ≼* 𝐴 | ||
Theorem | wdom2d 9617* | Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 5284). (Contributed by Stefan O'Rear, 13-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋) ⇒ ⊢ (𝜑 → 𝐴 ≼* 𝐵) | ||
Theorem | wdomd 9618* | Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.) |
⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋) ⇒ ⊢ (𝜑 → 𝐴 ≼* 𝐵) | ||
Theorem | brwdom3 9619* | Condition for weak dominance with a condition reminiscent of wdomd 9618. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋 ≼* 𝑌 ↔ ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦))) | ||
Theorem | brwdom3i 9620* | Weak dominance implies existence of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.) |
⊢ (𝑋 ≼* 𝑌 → ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)) | ||
Theorem | unwdomg 9621 | Weak dominance of a (disjoint) union. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷)) | ||
Theorem | xpwdomg 9622 | Weak dominance of a Cartesian product. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷)) | ||
Theorem | wdomima2g 9623 | A set is weakly dominant over its image under any function. This version of wdomimag 9624 is stated so as to avoid ax-rep 5284. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ≼* 𝐴) | ||
Theorem | wdomimag 9624 | A set is weakly dominant over its image under any function. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ≼* 𝐴) | ||
Theorem | unxpwdom2 9625 | Lemma for unxpwdom 9626. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝐴 × 𝐴) ≈ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)) | ||
Theorem | unxpwdom 9626 | If a Cartesian product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)) | ||
Theorem | ixpiunwdom 9627* | Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8966 this shows that ∪ 𝑥 ∈ 𝐴𝐵 and X𝑥 ∈ 𝐴𝐵 have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼* (X𝑥 ∈ 𝐴 𝐵 × 𝐴)) | ||
Theorem | harwdom 9628 | The value of the Hartogs function at a set 𝑋 is weakly dominated by 𝒫 (𝑋 × 𝑋). This follows from a more precise analysis of the bound used in hartogs 9581 to prove that (har‘𝑋) is an ordinal. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋)) | ||
Axiom | ax-reg 9629* | Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 9632) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 9633). A stronger version that works for proper classes is proved as zfregs 9769. (Contributed by NM, 14-Aug-1993.) |
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) | ||
Theorem | axreg2 9630* | Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.) |
⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) | ||
Theorem | zfregcl 9631* | The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.) |
⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴)) | ||
Theorem | zfreg 9632* | The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form". Axiom Reg of [BellMachover] p. 480. There is also a "strong form", not requiring that 𝐴 be a set, that can be proved with more difficulty (see zfregs 9769). (Contributed by NM, 26-Nov-1995.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅) | ||
Theorem | elirrv 9633 | The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr 9642 and efrirr 5668, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.) |
⊢ ¬ 𝑥 ∈ 𝑥 | ||
Theorem | elirr 9634 | No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. Theorem 1.9(i) of [Schloeder] p. 1. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ¬ 𝐴 ∈ 𝐴 | ||
Theorem | elneq 9635 | A class is not equal to any of its elements. (Contributed by AV, 14-Jun-2022.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) | ||
Theorem | nelaneq 9636 | A class is not an element of and equal to a class at the same time. Variant of elneq 9635 analogously to elnotel 9647 and en2lp 9643. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) |
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) | ||
Theorem | epinid0 9637 | The membership relation and the identity relation are disjoint. Variable-free version of nelaneq 9636. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) |
⊢ ( E ∩ I ) = ∅ | ||
Theorem | sucprcreg 9638 | A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.) |
⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴) | ||
Theorem | ruv 9639 | The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | ||
Theorem | ruALT 9640 | Alternate proof of ru 3788, simplified using (indirectly) the Axiom of Regularity ax-reg 9629. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Theorem | disjcsn 9641 | A class is disjoint from its singleton. A consequence of regularity. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 4-Apr-2019.) |
⊢ (𝐴 ∩ {𝐴}) = ∅ | ||
Theorem | zfregfr 9642 | The membership relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
⊢ E Fr 𝐴 | ||
Theorem | en2lp 9643 | No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) | ||
Theorem | elnanel 9644 | Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9643 and serves as an example in the context of Godel codes, see elnanelprv 35413. (Contributed by AV, 5-Nov-2023.) (New usage is discouraged.) |
⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) | ||
Theorem | cnvepnep 9645 | The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 9643. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 19-Jun-2022.) |
⊢ (◡ E ∩ E ) = ∅ | ||
Theorem | epnsym 9646 | The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.) |
⊢ ◡ E ≠ E | ||
Theorem | elnotel 9647 | A class cannot be an element of one of its elements. (Contributed by AV, 14-Jun-2022.) |
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ 𝐴) | ||
Theorem | elnel 9648 | A class cannot be an element of one of its elements. (Contributed by AV, 14-Jun-2022.) |
⊢ (𝐴 ∈ 𝐵 → 𝐵 ∉ 𝐴) | ||
Theorem | en3lplem1 9649* | Lemma for en3lp 9651. (Contributed by Alan Sare, 28-Oct-2011.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)) | ||
Theorem | en3lplem2 9650* | Lemma for en3lp 9651. (Contributed by Alan Sare, 28-Oct-2011.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)) | ||
Theorem | en3lp 9651 | No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 44842 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) |
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) | ||
Theorem | preleqg 9652 | Equality of two unordered pairs when one member of each pair contains the other member. Closed form of preleq 9653. (Contributed by AV, 15-Jun-2022.) |
⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | preleq 9653 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) (Revised by AV, 15-Jun-2022.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | preleqALT 9654 | Alternate proof of preleq 9653, not based on preleqg 9652: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐵 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | opthreg 9655 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 9629 (via the preleq 9653 step). See df-op 4637 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) (Proof shortened by AV, 15-Jun-2022.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | suc11reg 9656 | The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.) |
⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵) | ||
Theorem | dford2 9657* | Assuming ax-reg 9629, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.) |
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))) | ||
Theorem | inf0 9658* | Existence of ω implies our axiom of infinity ax-inf 9675. The proof shows that the especially contrived class "ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) " exists, is a subset of its union, and contains a given set 𝑥 (and thus is nonempty). Thus, it provides an example demonstrating that a set 𝑦 exists with the necessary properties demanded by ax-inf 9675. (Contributed by NM, 15-Oct-1996.) Revised to closed form. (Revised by BJ, 20-May-2024.) |
⊢ (ω ∈ 𝑉 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))) | ||
Theorem | inf1 9659 | Variation of Axiom of Infinity (using zfinf 9676 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.) |
⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) ⇒ ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) | ||
Theorem | inf2 9660* | Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 9676 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.) |
⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) ⇒ ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) | ||
Theorem | inf3lema 9661* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. (Contributed by NM, 28-Oct-1996.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ (𝐺‘𝐵) ↔ (𝐴 ∈ 𝑥 ∧ (𝐴 ∩ 𝑥) ⊆ 𝐵)) | ||
Theorem | inf3lemb 9662* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. (Contributed by NM, 28-Oct-1996.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹‘∅) = ∅ | ||
Theorem | inf3lemc 9663* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. (Contributed by NM, 28-Oct-1996.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ ω → (𝐹‘suc 𝐴) = (𝐺‘(𝐹‘𝐴))) | ||
Theorem | inf3lemd 9664* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. (Contributed by NM, 28-Oct-1996.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥) | ||
Theorem | inf3lem1 9665* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. (Contributed by NM, 28-Oct-1996.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴)) | ||
Theorem | inf3lem2 9666* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. (Contributed by NM, 28-Oct-1996.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ≠ 𝑥)) | ||
Theorem | inf3lem3 9667* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 9632. (Contributed by NM, 29-Oct-1996.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ≠ (𝐹‘suc 𝐴))) | ||
Theorem | inf3lem4 9668* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. (Contributed by NM, 29-Oct-1996.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ⊊ (𝐹‘suc 𝐴))) | ||
Theorem | inf3lem5 9669* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. (Contributed by NM, 29-Oct-1996.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))) | ||
Theorem | inf3lem6 9670* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. (Contributed by NM, 29-Oct-1996.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → 𝐹:ω–1-1→𝒫 𝑥) | ||
Theorem | inf3lem7 9671* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 7979. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.) |
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) & ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V) | ||
Theorem | inf3 9672 |
Our Axiom of Infinity ax-inf 9675 implies the standard Axiom of Infinity.
The hypothesis is a variant of our Axiom of Infinity provided by
inf2 9660, and the conclusion is the version of the Axiom of Infinity
shown as Axiom 7 in [TakeutiZaring] p. 43. (Other standard versions are
proved later as axinf2 9677 and zfinf2 9679.) The main proof is provided by
inf3lema 9661 through inf3lem7 9671, and this final piece eliminates the
auxiliary hypothesis of inf3lem7 9671. This proof is due to
Ian Sutherland, Richard Heck, and Norman Megill and was posted
on Usenet as shown below. Although the result is not new, the authors
were unable to find a published proof.
(As posted to sci.logic on 30-Oct-1996, with annotations added.) Theorem: The statement "There exists a nonempty set that is a subset of its union" implies the Axiom of Infinity. Proof: Let X be a nonempty set which is a subset of its union; the latter property is equivalent to saying that for any y in X, there exists a z in X such that y is in z. Define by finite recursion a function F:omega-->(power X) such that F_0 = 0 (See inf3lemb 9662.) F_n+1 = {y<X | y^X subset F_n} (See inf3lemc 9663.) Note: ^ means intersect, < means \in ("element of"). (Finite recursion as typically done requires the existence of omega; to avoid this we can just use transfinite recursion restricted to omega. F is a class-term that is not necessarily a set at this point.) Lemma 1. F_n subset F_n+1. (See inf3lem1 9665.) Proof: By induction: F_0 subset F_1. If y < F_n+1, then y^X subset F_n, so if F_n subset F_n+1, then y^X subset F_n+1, so y < F_n+2. Lemma 2. F_n =/= X. (See inf3lem2 9666.) Proof: By induction: F_0 =/= X because X is not empty. Assume F_n =/= X. Then there is a y in X that is not in F_n. By definition of X, there is a z in X that contains y. Suppose F_n+1 = X. Then z is in F_n+1, and z^X contains y, so z^X is not a subset of F_n, contrary to the definition of F_n+1. Lemma 3. F_n =/= F_n+1. (See inf3lem3 9667.) Proof: Using the identity y^X subset F_n <-> y^(X-F_n) = 0, we have F_n+1 = {y<X | y^(X-F_n) = 0}. Let q = {y<X-F_n | y^(X-F_n) = 0}. Then q subset F_n+1. Since X-F_n is not empty by Lemma 2 and q is the set of \in-minimal elements of X-F_n, by Foundation q is not empty, so q and therefore F_n+1 have an element not in F_n. Lemma 4. F_n proper_subset F_n+1. (See inf3lem4 9668.) Proof: Lemmas 1 and 3. Lemma 5. F_m proper_subset F_n, m < n. (See inf3lem5 9669.) Proof: Fix m and use induction on n > m. Basis: F_m proper_subset F_m+1 by Lemma 4. Induction: Assume F_m proper_subset F_n. Then since F_n proper_subset F_n+1, F_m proper_subset F_n+1 by transitivity of proper subset. By Lemma 5, F_m =/= F_n for m =/= n, so F is 1-1. (See inf3lem6 9670.) Thus, the inverse of F is a function with range omega and domain a subset of power X, so omega exists by Replacement. (See inf3lem7 9671.) Q.E.D.(Contributed by NM, 29-Oct-1996.) |
⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ⇒ ⊢ ω ∈ V | ||
Theorem | infeq5i 9673 | Half of infeq5 9674. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) | ||
Theorem | infeq5 9674 | The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 9680.) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 ↔ ω ∈ V) | ||
Axiom | ax-inf 9675* |
Axiom of Infinity. An axiom of Zermelo-Fraenkel set theory. This axiom
is the gateway to "Cantor's paradise" (an expression coined by
Hilbert).
It asserts that given a starting set 𝑥, an infinite set 𝑦 built
from it exists. Although our version is apparently not given in the
literature, it is similar to, but slightly shorter than, the Axiom of
Infinity in [FreydScedrov] p. 283
(see inf1 9659 and inf2 9660). More
standard versions, which essentially state that there exists a set
containing all the natural numbers, are shown as zfinf2 9679 and omex 9680 and
are based on the (nontrivial) proof of inf3 9672.
This version has the
advantage that when expanded to primitives, it has fewer symbols than
the standard version ax-inf2 9678. Theorem inf0 9658
shows the reverse
derivation of our axiom from a standard one. Theorem inf5 9682
shows a
very short way to state this axiom.
The standard version of Infinity ax-inf2 9678 requires this axiom along with Regularity ax-reg 9629 for its derivation (as Theorem axinf2 9677 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 9678 instead of this one. The derivation of this axiom from ax-inf2 9678 is shown by Theorem axinf 9681. Proofs should normally use the standard version ax-inf2 9678 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) | ||
Theorem | zfinf 9676* | Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.) |
⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) | ||
Theorem | axinf2 9677* |
A standard version of Axiom of Infinity, expanded to primitives, derived
from our version of Infinity ax-inf 9675 and Regularity ax-reg 9629.
This theorem should not be referenced in any proof. Instead, use ax-inf2 9678 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.) |
⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))) | ||
Axiom | ax-inf2 9678* | A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf2 9679 shows it converted to abbreviations. This axiom was derived as Theorem axinf2 9677 above, using our version of Infinity ax-inf 9675 and the Axiom of Regularity ax-reg 9629. We will reference ax-inf2 9678 instead of axinf2 9677 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 9675 from ax-inf2 9678 is shown by Theorem axinf 9681. (Contributed by NM, 3-Nov-1996.) |
⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))) | ||
Theorem | zfinf2 9679* | A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 9678 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.) |
⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) | ||
Theorem | omex 9680 |
The existence of omega (the class of natural numbers). Axiom 7 of
[TakeutiZaring] p. 43. Remark
1.21 of [Schloeder] p. 3. This theorem
is proved assuming the Axiom of Infinity and in fact is equivalent to
it, as shown by the reverse derivation inf0 9658.
A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial ¬ ω ∈ V; this would lead to ω = On by omon 7898 and Fin = V (the universe of all sets) by fineqv 9296. The finitist could still develop natural number, integer, and rational number arithmetic but would be denied the real numbers (as well as much of the rest of mathematics). In deference to the finitist, much of our development is done, when possible, without invoking the Axiom of Infinity; an example is Peano's axioms peano1 7910 through peano5 7915 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.) |
⊢ ω ∈ V | ||
Theorem | axinf 9681* | The first version of the Axiom of Infinity ax-inf 9675 proved from the second version ax-inf2 9678. Note that we didn't use ax-reg 9629, unlike the other direction axinf2 9677. (Contributed by NM, 24-Apr-2009.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) | ||
Theorem | inf5 9682 | The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see Theorem infeq5 9674). This provides us with a very compact way to express the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.) |
⊢ ∃𝑥 𝑥 ⊊ ∪ 𝑥 | ||
Theorem | omelon 9683 | Omega is an ordinal number. Theorem 1.22 of [Schloeder] p. 3. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
⊢ ω ∈ On | ||
Theorem | dfom3 9684* | The class of natural numbers ω can be defined as the intersection of all inductive sets (which is the smallest inductive set, since inductive sets are closed under intersection), which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. Definition 1.20 of [Schloeder] p. 3. (Contributed by NM, 6-Aug-1994.) |
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | ||
Theorem | elom3 9685* | A simplification of elom 7889 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.) |
⊢ (𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)) | ||
Theorem | dfom4 9686* | A simplification of df-om 7887 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.) |
⊢ ω = {𝑥 ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)} | ||
Theorem | dfom5 9687 | ω is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). Theorem 1.23 of [Schloeder] p. 4. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.) |
⊢ ω = ∩ {𝑥 ∣ Lim 𝑥} | ||
Theorem | oancom 9688 | Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.) |
⊢ (1o +o ω) ≠ (ω +o 1o) | ||
Theorem | isfinite 9689 | A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. The Axiom of Infinity is used for the forward implication. (Contributed by FL, 16-Apr-2011.) |
⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω) | ||
Theorem | fict 9690 | A finite set is countable (weaker version of isfinite 9689). (Contributed by Thierry Arnoux, 27-Mar-2018.) |
⊢ (𝐴 ∈ Fin → 𝐴 ≼ ω) | ||
Theorem | nnsdom 9691 | A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.) |
⊢ (𝐴 ∈ ω → 𝐴 ≺ ω) | ||
Theorem | omenps 9692 | Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.) |
⊢ ω ≈ (ω ∖ {∅}) | ||
Theorem | omensuc 9693 | The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.) |
⊢ ω ≈ suc ω | ||
Theorem | infdifsn 9694 | Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.) |
⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴) | ||
Theorem | infdiffi 9695 | Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ Fin) → (𝐴 ∖ 𝐵) ≈ 𝐴) | ||
Theorem | unbnn3 9696* | Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 9329 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.) |
⊢ ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≈ ω) | ||
Theorem | noinfep 9697* | Using the Axiom of Regularity in the form zfregfr 9642, show that there are no infinite descending ∈-chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.) |
⊢ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥) | ||
Syntax | ccnf 9698 | Extend class notation with the Cantor normal form function. |
class CNF | ||
Definition | df-cnf 9699* | Define the Cantor normal form function, which takes as input a finitely supported function from 𝑦 to 𝑥 and outputs the corresponding member of the ordinal exponential 𝑥 ↑o 𝑦. The content of the original Cantor Normal Form theorem is that for 𝑥 = ω this function is a bijection onto ω ↑o 𝑦 for any ordinal 𝑦 (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to On). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2 9732 of this function in terms of df-oi 9547. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
⊢ CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥 ↑m 𝑦) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ))) | ||
Theorem | cantnffval 9700* | The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) ⇒ ⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |