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Type | Label | Description |
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Statement | ||
Theorem | bj-vtoclg1fv 36901* | Version of bj-vtoclg1f 36900 with a disjoint variable condition on 𝑥, 𝑉. This removes dependency on df-sb 2062 and df-clab 2712. Prefer its use over bj-vtoclg1f 36900 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) & ⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓) | ||
Theorem | bj-vtoclg 36902* | A version of vtoclg 3553 with an additional disjoint variable condition (which is removable if we allow use of df-clab 2712, see bj-vtoclg1f 36900), which requires fewer axioms (i.e., removes dependency on ax-6 1964, ax-7 2004, ax-9 2115, ax-12 2174, ax-ext 2705, df-clab 2712, df-cleq 2726, df-v 3479). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) & ⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓) | ||
Theorem | bj-rabeqbid 36903 | Version of rabeqbidv 3451 with two disjoint variable conditions removed and the third replaced by a nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | bj-seex 36904* | Version of seex 5647 with a disjoint variable condition replaced by a nonfreeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ((𝑅 Se 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V) | ||
Theorem | bj-nfcf 36905* | Version of df-nfc 2889 with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 2-May-2019.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | bj-zfauscl 36906* |
General version of zfauscl 5303.
Remark: the comment in zfauscl 5303 is misleading: the essential use of ax-ext 2705 is the one via eleq2 2827 and not the one via vtocl 3557, since the latter can be proved without ax-ext 2705 (see bj-vtoclg 36902). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | ||
A few additional theorems on class abstractions and restricted class abstractions. | ||
Theorem | bj-elabd2ALT 36907* | Alternate proof of elabd2 3669 bypassing elab6g 3668 (and using sbiedvw 2092 instead of the ∀𝑥(𝑥 = 𝑦 → 𝜓) idiom). (Contributed by BJ, 16-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓}) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒)) | ||
Theorem | bj-unrab 36908* | Generalization of unrab 4320. Equality need not hold. (Contributed by BJ, 21-Apr-2019.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜓}) ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} | ||
Theorem | bj-inrab 36909 | Generalization of inrab 4321. (Contributed by BJ, 21-Apr-2019.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐵 ∣ 𝜓}) = {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ (𝜑 ∧ 𝜓)} | ||
Theorem | bj-inrab2 36910 | Shorter proof of inrab 4321. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} | ||
Theorem | bj-inrab3 36911* | Generalization of dfrab3ss 4328, which it may shorten. (Contributed by BJ, 21-Apr-2019.) (Revised by OpenAI, 7-Jul-2020.) |
⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) | ||
Theorem | bj-rabtr 36912* | Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.) |
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 | ||
Theorem | bj-rabtrALT 36913* | Alternate proof of bj-rabtr 36912. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 | ||
Theorem | bj-rabtrAUTO 36914* | Proof of bj-rabtr 36912 found automatically by the Metamath program "MM-PA> IMPROVE ALL / DEPTH 3 / 3" command followed by "MM-PA> MINIMIZE_WITH *". (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 | ||
Syntax | bj-cgab 36915 | Syntax for generalized class abstractions. |
class {𝐴 ∣ 𝑥 ∣ 𝜑} | ||
Definition | df-bj-gab 36916* | Definition of generalized class abstractions: typically, 𝑥 is a bound variable in 𝐴 and 𝜑 and {𝐴 ∣ 𝑥 ∣ 𝜑} denotes "the class of 𝐴(𝑥)'s such that 𝜑(𝑥)". (Contributed by BJ, 4-Oct-2024.) |
⊢ {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)} | ||
Theorem | bj-gabss 36917 | Inclusion of generalized class abstractions. (Contributed by BJ, 4-Oct-2024.) |
⊢ (∀𝑥(𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → {𝐴 ∣ 𝑥 ∣ 𝜑} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜓}) | ||
Theorem | bj-gabssd 36918 | Inclusion of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒}) | ||
Theorem | bj-gabeqd 36919 | Equality of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} = {𝐵 ∣ 𝑥 ∣ 𝜒}) | ||
Theorem | bj-gabeqis 36920* | Equality of generalized class abstractions, with implicit substitution. (Contributed by BJ, 4-Oct-2024.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝐵 ∣ 𝑦 ∣ 𝜓} | ||
Theorem | bj-elgab 36921 | Elements of a generalized class abstraction. (Contributed by BJ, 4-Oct-2024.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → (∃𝑥(𝐴 = 𝐵 ∧ 𝜓) ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝐴 ∈ {𝐵 ∣ 𝑥 ∣ 𝜓} ↔ 𝜒)) | ||
Theorem | bj-gabima 36922 |
Generalized class abstraction as a direct image.
TODO: improve the support lemmas elimag 6083 and fvelima 6973 to nonfreeness hypothesis (and for the latter, biconditional). (Contributed by BJ, 4-Oct-2024.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝐹) & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ dom 𝐹) ⇒ ⊢ (𝜑 → {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} = (𝐹 “ {𝑥 ∣ 𝜓})) | ||
In this subsection, we define restricted nonfreeness (or relative nonfreeness). | ||
Syntax | wrnf 36923 | Syntax for restricted nonfreeness. |
wff Ⅎ𝑥 ∈ 𝐴𝜑 | ||
Definition | df-bj-rnf 36924 | Definition of restricted nonfreeness. Informally, the proposition Ⅎ𝑥 ∈ 𝐴𝜑 means that 𝜑(𝑥) does not vary on 𝐴. (Contributed by BJ, 19-Mar-2021.) |
⊢ (Ⅎ𝑥 ∈ 𝐴𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)) | ||
A few results around Russell's paradox. For clarity, we prove separately a FOL statement (now in the main part as ru0 2124) and then two versions (bj-ru1 36925 and bj-ru 36926). Special attention is put on minimizing axiom depencencies. | ||
Theorem | bj-ru1 36925* | A version of Russell's paradox ru 3788 not mentioning the universal class. (see also bj-ru 36926). (Contributed by BJ, 12-Oct-2019.) Remove usage of ax-10 2138, ax-11 2154, ax-12 2174 by using eqabbw 2812 following BTernaryTau's similar revision of ru 3788. (Revised by BJ, 28-Jun-2025.) (Proof modification is discouraged.) |
⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} | ||
Theorem | bj-ru 36926 | Remove dependency on ax-13 2374 (and df-v 3479) from Russell's paradox ru 3788 expressed with primitive symbols and with a class variable 𝑉. Note the more economical use of elissetv 2819 instead of isset 3491 to avoid use of df-v 3479. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) |
⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉 | ||
Theorem | currysetlem 36927* | Lemma for currysetlem 36927, where it is used with (𝑥 ∈ 𝑥 → 𝜑) substituted for 𝜓. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) |
⊢ ({𝑥 ∣ 𝜓} ∈ 𝑉 → ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑))) | ||
Theorem | curryset 36928* | Curry's paradox in set theory. This can be seen as a generalization of Russell's paradox, which corresponds to the case where 𝜑 is ⊥. See alternate exposal of basically the same proof currysetALT 36932. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) |
⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉 | ||
Theorem | currysetlem1 36929* | Lemma for currysetALT 36932. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) |
⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ⇒ ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑))) | ||
Theorem | currysetlem2 36930* | Lemma for currysetALT 36932. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) |
⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ⇒ ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑)) | ||
Theorem | currysetlem3 36931* | Lemma for currysetALT 36932. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) |
⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ⇒ ⊢ ¬ 𝑋 ∈ 𝑉 | ||
Theorem | currysetALT 36932* | Alternate proof of curryset 36928, or more precisely alternate exposal of the same proof. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉 | ||
A few utility theorems on disjointness of classes. | ||
Theorem | bj-n0i 36933* | Inference associated with n0 4358. Shortens 2ndcdisj 23479 (2888>2878), notzfaus 5368 (264>253). (Contributed by BJ, 22-Apr-2019.) |
⊢ 𝐴 ≠ ∅ ⇒ ⊢ ∃𝑥 𝑥 ∈ 𝐴 | ||
Theorem | bj-disjsn01 36934 | Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9641 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.) |
⊢ ({∅} ∩ {1o}) = ∅ | ||
Theorem | bj-0nel1 36935 | The empty set does not belong to {1o}. (Contributed by BJ, 6-Apr-2019.) |
⊢ ∅ ∉ {1o} | ||
Theorem | bj-1nel0 36936 | 1o does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.) |
⊢ 1o ∉ {∅} | ||
A few utility theorems on direct products. | ||
Theorem | bj-xpimasn 36937 | The image of a singleton, general case. [Change and relabel xpimasn 6206 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.) |
⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅) | ||
Theorem | bj-xpima1sn 36938 | The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 6206 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.) |
⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅) | ||
Theorem | bj-xpima1snALT 36939 | Alternate proof of bj-xpima1sn 36938. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅) | ||
Theorem | bj-xpima2sn 36940 | The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6206.] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) |
⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) | ||
Theorem | bj-xpnzex 36941 | If the first factor of a product is nonempty, and the product is a set, then the second factor is a set. UPDATE: this is actually the curried (exported) form of xpexcnv 7942 (up to commutation in the product). (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.) |
⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝑉 → 𝐵 ∈ V)) | ||
Theorem | bj-xpexg2 36942 | Curried (exported) form of xpexg 7768. (Contributed by BJ, 2-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × 𝐵) ∈ V)) | ||
Theorem | bj-xpnzexb 36943 | If the first factor of a product is a nonempty set, then the product is a set if and only if the second factor is a set. (Contributed by BJ, 2-Apr-2019.) |
⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V ↔ (𝐴 × 𝐵) ∈ V)) | ||
Theorem | bj-cleq 36944* | Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.) |
⊢ (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)}) | ||
This subsection introduces the "singletonization" and the "tagging" of a class. The singletonization of a class is the class of singletons of elements of that class. It is useful since all nonsingletons are disjoint from it, so one can easily adjoin to it disjoint elements, which is what the tagging does: it adjoins the empty set. This can be used for instance to define the one-point compactification of a topological space. It will be used in the next section to define tuples which work for proper classes. | ||
Theorem | bj-snsetex 36945* | The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5284. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) | ||
Theorem | bj-clexab 36946* | Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐵)} ∈ V) | ||
Syntax | bj-csngl 36947 | Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.) |
class sngl 𝐴 | ||
Definition | df-bj-sngl 36948* | Definition of "singletonization". The class sngl 𝐴 is isomorphic to 𝐴 and since it contains only singletons, it can be easily be adjoined disjoint elements, which can be useful in various constructions. (Contributed by BJ, 6-Oct-2018.) |
⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}} | ||
Theorem | bj-sngleq 36949 | Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵) | ||
Theorem | bj-elsngl 36950* | Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥}) | ||
Theorem | bj-snglc 36951 | Characterization of the elements of 𝐴 in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵) | ||
Theorem | bj-snglss 36952 | The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.) |
⊢ sngl 𝐴 ⊆ 𝒫 𝐴 | ||
Theorem | bj-0nelsngl 36953 | The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 8504). (Contributed by BJ, 6-Oct-2018.) |
⊢ ∅ ∉ sngl 𝐴 | ||
Theorem | bj-snglinv 36954* | Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.) |
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴} | ||
Theorem | bj-snglex 36955 | A class is a set if and only if its singletonization is a set. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) | ||
Syntax | bj-ctag 36956 | Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.) |
class tag 𝐴 | ||
Definition | df-bj-tag 36957 | Definition of the tagged copy of a class, that is, the adjunction to (an isomorph of) 𝐴 of a disjoint element (here, the empty set). Remark: this could be used for the one-point compactification of a topological space. (Contributed by BJ, 6-Oct-2018.) |
⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | ||
Theorem | bj-tageq 36958 | Substitution property for tag. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵) | ||
Theorem | bj-eltag 36959* | Characterization of the elements of the tagging of a class. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ tag 𝐵 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅)) | ||
Theorem | bj-0eltag 36960 | The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.) |
⊢ ∅ ∈ tag 𝐴 | ||
Theorem | bj-tagn0 36961 | The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.) |
⊢ tag 𝐴 ≠ ∅ | ||
Theorem | bj-tagss 36962 | The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.) |
⊢ tag 𝐴 ⊆ 𝒫 𝐴 | ||
Theorem | bj-snglsstag 36963 | The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.) |
⊢ sngl 𝐴 ⊆ tag 𝐴 | ||
Theorem | bj-sngltagi 36964 | The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ sngl 𝐵 → 𝐴 ∈ tag 𝐵) | ||
Theorem | bj-sngltag 36965 | The singletonization and the tagging of a set contain the same singletons. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵)) | ||
Theorem | bj-tagci 36966 | Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ tag 𝐵) | ||
Theorem | bj-tagcg 36967 | Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ tag 𝐵)) | ||
Theorem | bj-taginv 36968* | Inverse of tagging. (Contributed by BJ, 6-Oct-2018.) |
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴} | ||
Theorem | bj-tagex 36969 | A class is a set if and only if its tagging is a set. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ V ↔ tag 𝐴 ∈ V) | ||
Theorem | bj-xtageq 36970 | The products of a given class and the tagging of either of two equal classes are equal. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐴 = 𝐵 → (𝐶 × tag 𝐴) = (𝐶 × tag 𝐵)) | ||
Theorem | bj-xtagex 36971 | The product of a set and the tagging of a set is a set. (Contributed by BJ, 2-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × tag 𝐵) ∈ V)) | ||
This subsection gives a definition of an ordered pair, or couple (2-tuple), that "works" for proper classes, as evidenced by Theorems bj-2uplth 37003 and bj-2uplex 37004, and more importantly, bj-pr21val 36995 and bj-pr22val 37001. In particular, one can define well-behaved tuples of classes. Classes in ZF(C) are only virtual, and in particular they cannot be quantified over. Theorem bj-2uplex 37004 has advantages: in view of df-br 5148, several sethood antecedents could be removed from existing theorems. For instance, relsnopg 5815 (resp. relsnop 5817) would hold without antecedents (resp. hypotheses) thanks to relsnb 5814). Also, the antecedent Rel 𝑅 could be removed from brrelex12 5740 and related theorems brrelex*, and, as a consequence, of multiple later theorems. Similarly, df-struct 17180 could be simplified by removing the exception currently made for the empty set. The projections are denoted by pr1 and pr2 and the couple with projections (or coordinates) 𝐴 and 𝐵 is denoted by ⦅𝐴, 𝐵⦆. Note that this definition uses the Kuratowski definition (df-op 4637) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 9655) without needing the axiom of regularity; it could even bypass this definition by "inlining" it. This definition is due to Anthony Morse and is expounded (with idiosyncratic notation) in Anthony P. Morse, A Theory of Sets, Academic Press, 1965 (second edition 1986). Note that this extends in a natural way to tuples. A variation of this definition is justified in opthprc 5752, but here we use "tagged versions" of the factors (see df-bj-tag 36957) so that an m-tuple can equal an n-tuple only when m = n (and the projections are the same). A comparison of the different definitions of tuples (strangely not mentioning Morse's), is given in Dominic McCarty and Dana Scott, Reconsidering ordered pairs, Bull. Symbolic Logic, Volume 14, Issue 3 (Sept. 2008), 379--397. where a recursive definition of tuples is given that avoids the two-step definition of tuples and that can be adapted to various set theories. Finally, another survey is Akihiro Kanamori, The empty set, the singleton, and the ordered pair, Bull. Symbolic Logic, Volume 9, Number 3 (Sept. 2003), 273--298. (available at http://math.bu.edu/people/aki/8.pdf 36957) | ||
Syntax | bj-cproj 36972 | Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.) |
class (𝐴 Proj 𝐵) | ||
Definition | df-bj-proj 36973* | Definition of the class projection corresponding to tagged tuples. The expression (𝐴 Proj 𝐵) denotes the projection on the A^th component. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.) |
⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} | ||
Theorem | bj-projeq 36974 | Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷))) | ||
Theorem | bj-projeq2 36975 | Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶)) | ||
Theorem | bj-projun 36976 | The class projection on a given component preserves unions. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)) | ||
Theorem | bj-projex 36977 | Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V) | ||
Theorem | bj-projval 36978 | Value of the class projection. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅)) | ||
Syntax | bj-c1upl 36979 | Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.) |
class ⦅𝐴⦆ | ||
Definition | df-bj-1upl 36980 | Definition of the Morse monuple (1-tuple). This is not useful per se, but is used as a step towards the definition of couples (2-tuples, or ordered pairs). The reason for "tagging" the set is so that an m-tuple and an n-tuple be equal only when m = n. Note that with this definition, the 0-tuple is the empty set. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 36994, bj-2uplth 37003, bj-2uplex 37004, and the properties of the projections (see df-bj-pr1 36983 and df-bj-pr2 36997). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.) |
⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | ||
Theorem | bj-1upleq 36981 | Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆) | ||
Syntax | bj-cpr1 36982 | Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.) |
class pr1 𝐴 | ||
Definition | df-bj-pr1 36983 | Definition of the first projection of a class tuple. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-pr1eq 36984, bj-pr11val 36987, bj-pr21val 36995, bj-pr1ex 36988. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.) |
⊢ pr1 𝐴 = (∅ Proj 𝐴) | ||
Theorem | bj-pr1eq 36984 | Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵) | ||
Theorem | bj-pr1un 36985 | The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.) |
⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵) | ||
Theorem | bj-pr1val 36986 | Value of the first projection. (Contributed by BJ, 6-Apr-2019.) |
⊢ pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅) | ||
Theorem | bj-pr11val 36987 | Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.) |
⊢ pr1 ⦅𝐴⦆ = 𝐴 | ||
Theorem | bj-pr1ex 36988 | Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 ∈ 𝑉 → pr1 𝐴 ∈ V) | ||
Theorem | bj-1uplth 36989 | The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.) |
⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵) | ||
Theorem | bj-1uplex 36990 | A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.) |
⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) | ||
Theorem | bj-1upln0 36991 | A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.) |
⊢ ⦅𝐴⦆ ≠ ∅ | ||
Syntax | bj-c2uple 36992 | Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.) |
class ⦅𝐴, 𝐵⦆ | ||
Definition | df-bj-2upl 36993 | Definition of the Morse couple. See df-bj-1upl 36980. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 36994, bj-2uplth 37003, bj-2uplex 37004, and the properties of the projections (see df-bj-pr1 36983 and df-bj-pr2 36997). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.) |
⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | ||
Theorem | bj-2upleq 36994 | Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆)) | ||
Theorem | bj-pr21val 36995 | Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.) |
⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 | ||
Syntax | bj-cpr2 36996 | Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.) |
class pr2 𝐴 | ||
Definition | df-bj-pr2 36997 | Definition of the second projection of a class tuple. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-pr2eq 36998, bj-pr22val 37001, bj-pr2ex 37002. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.) |
⊢ pr2 𝐴 = (1o Proj 𝐴) | ||
Theorem | bj-pr2eq 36998 | Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.) |
⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵) | ||
Theorem | bj-pr2un 36999 | The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.) |
⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵) | ||
Theorem | bj-pr2val 37000 | Value of the second projection. (Contributed by BJ, 6-Apr-2019.) |
⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅) |
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