Home Metamath Proof ExplorerTheorem List (p. 331 of 429) < 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: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

Theorem List for Metamath Proof Explorer - 33001-33100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-ceqsaltv 33001* Version of bj-ceqsalt 33000 with a dv condition on 𝑥, 𝑉, removing dependency on df-sb 1938 and df-clab 2638. Prefer its use over bj-ceqsalt 33000 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theorembj-ceqsalg0 33002 The FOL content of ceqsalg 3261. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒 → (𝜑𝜓))       (∃𝑥𝜒 → (∀𝑥(𝜒𝜑) ↔ 𝜓))

Theorembj-ceqsalg 33003* Remove from ceqsalg 3261 dependency on ax-ext 2631 (and on df-cleq 2644 and df-v 3233). See also bj-ceqsalgv 33005. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theorembj-ceqsalgALT 33004* Alternate proof of bj-ceqsalg 33003. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theorembj-ceqsalgv 33005* Version of bj-ceqsalg 33003 with a dv condition on 𝑥, 𝑉, removing dependency on df-sb 1938 and df-clab 2638. Prefer its use over bj-ceqsalg 33003 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theorembj-ceqsalgvALT 33006* Alternate proof of bj-ceqsalgv 33005. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theorembj-ceqsal 33007* Remove from ceqsal 3263 dependency on ax-ext 2631 (and on df-cleq 2644, df-v 3233, df-clab 2638, df-sb 1938). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)

Theorembj-ceqsalv 33008* Remove from ceqsalv 3264 dependency on ax-ext 2631 (and on df-cleq 2644, df-v 3233, df-clab 2638, df-sb 1938). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)

Theorembj-spcimdv 33009* Remove from spcimdv 3321 dependency on ax-9 2039, ax-10 2059, ax-11 2074, ax-13 2282, ax-ext 2631, df-cleq 2644 (and df-nfc 2782, df-v 3233, df-or 384, df-tru 1526, df-nf 1750). For an even more economical version, see bj-spcimdvv 33010. (Contributed by BJ, 30-Nov-2020.) (Proof modification is discouraged.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))

Theorembj-spcimdvv 33010* Remove from spcimdv 3321 dependency on ax-7 1981, ax-8 2032, ax-10 2059, ax-11 2074, ax-12 2087 ax-13 2282, ax-ext 2631, df-cleq 2644, df-clab 2638 (and df-nfc 2782, df-v 3233, df-or 384, df-tru 1526, df-nf 1750) at the price of adding a DV condition on 𝑥, 𝐵 (but in usages, 𝑥 is typically a dummy, hence fresh, variable). For the version without this DV condition, see bj-spcimdv 33009. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))

20.14.5.3  The class-form not-free predicate

In this section, we prove the symmetry of the class-form not-free predicate.

Theorembj-nfcsym 33011 The class-form not-free predicate defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 4927 with additional axioms; see also nfcv 2793). This could be proved from aecom 2344 and nfcvb 4928 but the latter requires a domain with at least two objects (hence uses extra axioms). (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid use of eqcomd 2657 instead of equcomd 1992; removing dependency on ax-ext 2631 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2811, eleq2d 2716 (using elequ2 2044), nfcvf 2817, dvelimc 2816, dvelimdc 2815, nfcvf2 2818. (Proof modification is discouraged.)
(𝑥𝑦𝑦𝑥)

20.14.5.4  Proposal for the definitions of class membership and class equality

In this section, we show (bj-ax8 33012 and bj-ax9 33015) that the current forms of the definitions of class membership (df-clel 2647) and class equality (df-cleq 2644) are too powerful, and we propose alternate definitions (bj-df-clel 33013 and bj-df-cleq 33018) which also have the advantage of making it clear that these definitions are conservative.

Theorembj-ax8 33012 Proof of ax-8 2032 from df-clel 2647 (and FOL). This shows that df-clel 2647 is "too powerful". A possible definition is given by bj-df-clel 33013. (Contributed by BJ, 27-Jun-2019.) Also a direct consequence of eleq1w 2713, which has essentially the same proof. (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theorembj-df-clel 33013* Candidate definition for df-clel 2647 (the need for it is exposed in bj-ax8 33012). The similarity of the hypothesis and the conclusion, together with all possible dv conditions, makes it clear that this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This definition should be directly referenced only by bj-dfclel 33014, which should be used instead. The proof is irrelevant since this is a proposal for an axiom.

Note: the current definition df-clel 2647 already mentions cleljust 2038 as a justification; here, we merely propose to put it (more preciesly: its universal closure) as a hypothesis to make things more explicit. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)

𝑢𝑣(𝑢𝑣 ↔ ∃𝑤(𝑤 = 𝑢𝑤𝑣))       (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))

Theorembj-dfclel 33014* Characterization of the elements of a class. Note: cleljust 2038 could be relabeled "clelhyp". (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)
(𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))

Theorembj-ax9 33015* Proof of ax-9 2039 from Tarski's FOL=, sp 2091, df-cleq 2644 and ax-ext 2631 (with two extra dv conditions on 𝑥, 𝑧 and 𝑦, 𝑧). For a version without these dv conditions, see bj-ax9-2 33016. This shows that df-cleq 2644 is "too powerful". A possible definition is given by bj-df-cleq 33018. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theorembj-ax9-2 33016 Proof of ax-9 2039 from Tarski's FOL=, ax-8 2032 (specifically, ax8v1 2034 and ax8v2 2035) , df-cleq 2644 and ax-ext 2631. For a version not using ax-8 2032, see bj-ax9 33015. This shows that df-cleq 2644 is "too powerful". A possible definition is given by bj-df-cleq 33018. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theorembj-cleqhyp 33017* The hypothesis of bj-df-cleq 33018. Note that the hypothesis of bj-df-cleq 33018 actually has an additional dv condition on 𝑥, 𝑦 and therefore is provable by simply using ax-ext 2631 in place of axext3 2633 in the current proof. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))

Theorembj-df-cleq 33018* Candidate definition for df-cleq 2644 (the need for it is exposed in bj-ax9 33015). The similarity of the hypothesis and the conclusion makes it clear that this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This definition should be directly referenced only by bj-dfcleq 33019, which should be used instead. The proof is irrelevant since this is a proposal for an axiom.

Another definition, which would give finer control, is actually the pair of definitions where each has one implication of the present biconditional as hypothesis and conclusion. They assert that extensionality (respectively, the left-substitution axiom for the membership predicate) extends from setvars to classes. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

𝑢𝑣(𝑢 = 𝑣 ↔ ∀𝑤(𝑤𝑢𝑤𝑣))       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theorembj-dfcleq 33019* Proof of class extensionality from the axiom of set extensionality (ax-ext 2631) and the definition of class equality (bj-df-cleq 33018). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

20.14.5.5  Lemmas for class substitution

Some useful theorems for dealing with substitutions: sbbi 2429, sbcbig 3513, sbcel1g 4020, sbcel2 4022, sbcel12 4016, sbceqg 4017, csbvarg 4036.

Theorembj-sbeqALT 33020* Substitution in an equality (use the more genereal version bj-sbeq 33021 instead, without disjoint variable condition). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)

Theorembj-sbeq 33021 Distribute proper substitution through an equality relation. (See sbceqg 4017). (Contributed by BJ, 6-Oct-2018.)
([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)

Theorembj-sbceqgALT 33022 Distribute proper substitution through an equality relation. Alternate proof of sbceqg 4017. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 4017, but "minimize */except sbceqg" is ok. (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

Theorembj-csbsnlem 33023* Lemma for bj-csbsn 33024 (in this lemma, 𝑥 cannot occur in 𝐴). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
𝐴 / 𝑥{𝑥} = {𝐴}

Theorembj-csbsn 33024 Substitution in a singleton. (Contributed by BJ, 6-Oct-2018.)
𝐴 / 𝑥{𝑥} = {𝐴}

Theorembj-sbel1 33025* Version of sbcel1g 4020 when substituting a set. (Note: one could have a corresponding version of sbcel12 4016 when substituting a set, but the point here is that the antecedent of sbcel1g 4020 is not needed when substituting a set.) (Contributed by BJ, 6-Oct-2018.)
([𝑦 / 𝑥]𝐴𝐵𝑦 / 𝑥𝐴𝐵)

Theorembj-abv 33026 The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
(∀𝑥𝜑 → {𝑥𝜑} = V)

Theorembj-ab0 33027 The class of sets verifying a falsity is the empty set (closed form of abf 4011). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
(∀𝑥 ¬ 𝜑 → {𝑥𝜑} = ∅)

Theorembj-abf 33028 Shorter proof of abf 4011 (which should be kept as abfALT). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
¬ 𝜑       {𝑥𝜑} = ∅

Theorembj-csbprc 33029 More direct proof of csbprc 4013 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)

20.14.5.6  Removing some dv conditions

Theorembj-exlimmpi 33030 Lemma for bj-vtoclg1f1 33035 (an instance of this lemma is a version of bj-vtoclg1f1 33035 where 𝑥 and 𝑦 are identified). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒 → (𝜑𝜓))    &   𝜑       (∃𝑥𝜒𝜓)

Theorembj-exlimmpbi 33031 Lemma for theorems of the vtoclg 3297 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒 → (𝜑𝜓))    &   𝜑       (∃𝑥𝜒𝜓)

Theorembj-exlimmpbir 33032 Lemma for theorems of the vtoclg 3297 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜒 → (𝜑𝜓))    &   𝜓       (∃𝑥𝜒𝜑)

Theorembj-vtoclf 33033* Remove dependency on ax-ext 2631, df-clab 2638 and df-cleq 2644 (and df-sb 1938 and df-v 3233) from vtoclf 3289. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   𝐴𝑉    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓

Theorembj-vtocl 33034* Remove dependency on ax-ext 2631, df-clab 2638 and df-cleq 2644 (and df-sb 1938 and df-v 3233) from vtocl 3290. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝐴𝑉    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓

Theorembj-vtoclg1f1 33035* The FOL content of vtoclg1f 3296 (hence not using ax-ext 2631, df-cleq 2644, df-nfc 2782, df-v 3233). Note the weakened "major" hypothesis and the dv condition between 𝑥 and 𝐴 (needed since the class-form non-free predicate is not available without ax-ext 2631; as a byproduct, this dispenses with ax-11 2074 and ax-13 2282). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (∃𝑦 𝑦 = 𝐴𝜓)

Theorembj-vtoclg1f 33036* Reprove vtoclg1f 3296 from bj-vtoclg1f1 33035. This removes dependency on ax-ext 2631, df-cleq 2644 and df-v 3233. Use bj-vtoclg1fv 33037 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theorembj-vtoclg1fv 33037* Version of bj-vtoclg1f 33036 with a dv condition on 𝑥, 𝑉. This removes dependency on df-sb 1938 and df-clab 2638. Prefer its use over bj-vtoclg1f 33036 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theorembj-rabbida2 33038 Version of rabbidva2 3217 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theorembj-rabbida 33039 Version of rabbidva 3219 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theorembj-rabbid 33040 Version of rabbidv 3220 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theorembj-rabeqd 33041 Deduction form of rabeq 3223. Note that contrary to rabeq 3223 it has no dv condition. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})

Theorembj-rabeqbid 33042 Version of rabeqbidv 3226 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theorembj-rabeqbida 33043 Version of rabeqbidva 3227 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theorembj-seex 33044* Version of seex 5106 with a dv condition replaced by a non-freeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.)
𝑥𝐵       ((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)

Theorembj-nfcf 33045* Version of df-nfc 2782 with a dv condition replaced with a non-freeness hypothesis. (Contributed by BJ, 2-May-2019.)
𝑦𝐴       (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)

Theorembj-axsep2 33046* Remove dependency on ax-12 2087 and ax-13 2282 from axsep2 4815 while shortening its proof. Remark: the comment in zfauscl 4816 is misleading: the essential use of ax-ext 2631 is the one via eleq2 2719 and not the one via vtocl 3290, since the latter can be proved without ax-ext 2631 (see bj-vtocl 33034). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

20.14.5.7  Class abstractions

A few additional theorems on class abstractions and restricted class abstractions.

Theorembj-unrab 33047* Generalization of unrab 3931. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)
({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜓}) ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}

Theorembj-inrab 33048 Generalization of inrab 3932. (Contributed by BJ, 21-Apr-2019.)
({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}

Theorembj-inrab2 33049 Shorter proof of inrab 3932. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Theorembj-inrab3 33050* Generalization of dfrab3ss 3938, which it may shorten. (Contributed by BJ, 21-Apr-2019.) (Revised by OpenAI, 7-Jul-2020.)
(𝐴 ∩ {𝑥𝐵𝜑}) = ({𝑥𝐴𝜑} ∩ 𝐵)

Theorembj-rabtr 33051* Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.)
{𝑥𝐴 ∣ ⊤} = 𝐴

Theorembj-rabtrALT 33052* Alternate proof of bj-rabtr 33051. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝐴 ∣ ⊤} = 𝐴

Theorembj-rabtrALTALT 33053* Alternate proof of bj-rabtr 33051. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝐴 ∣ ⊤} = 𝐴

Theorembj-rabtrAUTO 33054* Proof of bj-rabtr 33051 found automatically by "improve all /depth 3 /3" followed by "minimize *". (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝐴 ∣ ⊤} = 𝐴

20.14.5.8  Restricted non-freeness

In this subsection, we define restricted non-freeness (or relative non-freeness).

Syntaxwrnf 33055 Syntax for restricted non-freeness.
wff 𝑥𝐴𝜑

Definitiondf-bj-rnf 33056 Definition of restricted non-freeness. Informally, the proposition 𝑥𝐴𝜑 means that 𝜑(𝑥) does not vary on 𝐴. (Contributed by BJ, 19-Mar-2021.)
(Ⅎ𝑥𝐴𝜑 ↔ (∃𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜑))

A few results around Russell's paradox. For clarity, we prove separately its FOL part (bj-ru0 33057) and then two versions (bj-ru1 33058 and bj-ru 33059). Special attention is put on minimizing axiom depencencies.

Theorembj-ru0 33057* The FOL part of Russell's paradox ru 3467 (see also bj-ru1 33058, bj-ru 33059). Use of elequ1 2037, bj-elequ12 32793, bj-spvv 32848 (instead of eleq1 2718, eleq12d 2724, spv 2296 as in ru 3467) permits to remove dependency on ax-10 2059, ax-11 2074, ax-12 2087, ax-13 2282, ax-ext 2631, df-sb 1938, df-clab 2638, df-cleq 2644, df-clel 2647. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥)

Theorembj-ru1 33058* A version of Russell's paradox ru 3467 (see also bj-ru 33059). Note the more economical use of bj-abeq2 32898 instead of abeq2 2761 to avoid dependency on ax-13 2282. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}

Theorembj-ru 33059 Remove dependency on ax-13 2282 (and df-v 3233) from Russell's paradox ru 3467 expressed with primitive symbols and with a class variable 𝑉 (note that axsep2 4815 does require ax-8 2032 and ax-9 2039 since it requires df-clel 2647 and df-cleq 2644--- see bj-df-clel 33013 and bj-df-cleq 33018). Note the more economical use of bj-elissetv 32986 instead of isset 3238 to avoid use of df-v 3233. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ {𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉

20.14.5.10  Some disjointness results

A few utility theorems on disjointness of classes.

Theorembj-n0i 33060* Inference associated with n0 3964. Shortens 2ndcdisj 21307 (2888>2878), notzfaus 4870 (264>253). (Contributed by BJ, 22-Apr-2019.)
𝐴 ≠ ∅       𝑥 𝑥𝐴

Theorembj-disjcsn 33061 A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 30931 and does not depend on df-ne 2824. (Contributed by BJ, 4-Apr-2019.)
(𝐴 ∩ {𝐴}) = ∅

Theorembj-disjsn01 33062 Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 33061 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)
({∅} ∩ {1𝑜}) = ∅

Theorembj-1ex 33063 1𝑜 is a set. (Contributed by BJ, 6-Apr-2019.)
1𝑜 ∈ V

Theorembj-2ex 33064 2𝑜 is a set. (Contributed by BJ, 6-Apr-2019.)
2𝑜 ∈ V

Theorembj-0nel1 33065 The empty set does not belong to {1𝑜}. (Contributed by BJ, 6-Apr-2019.)
∅ ∉ {1𝑜}

Theorembj-1nel0 33066 1𝑜 does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.)
1𝑜 ∉ {∅}

20.14.5.11  Complements on direct products

A few utility theorems on direct products.

Theorembj-xpimasn 33067 The image of a singleton, general case. [Change and relabel xpimasn 5614 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)

Theorembj-xpima1sn 33068 The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 5614 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Theorembj-xpima1snALT 33069 Alternate proof of bj-xpima1sn 33068. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Theorembj-xpima2sn 33070 The image of a singleton by a direct product, nonempty case. [To replace xpimasn 5614] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Theorembj-xpnzex 33071 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 7150 (up to commutation in the product). (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
(𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝑉𝐵 ∈ V))

Theorembj-xpexg2 33072 Curried (exported) form of xpexg 7002. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → (𝐵𝑊 → (𝐴 × 𝐵) ∈ V))

Theorembj-xpnzexb 33073 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))

Theorembj-cleq 33074* Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)
(𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})

20.14.5.12  "Singletonization" and tagging

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.

Theorembj-sels 33075* If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.)
(𝐴𝑉 → ∃𝑥 𝐴𝑥)

Theorembj-snsetex 33076* The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 4804. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)

Theorembj-clex 33077* Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐵)} ∈ V)

Syntaxbj-csngl 33078 Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
class sngl 𝐴

Definitiondf-bj-sngl 33079* Definition of "singletonization". The class sngl 𝐴 is isomorphic to 𝐴 and since it contains only singletons, it can be adjoined disjoint elements, which can be useful in various constructions. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥 = {𝑦}}

Theorembj-sngleq 33080 Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)

Theorembj-elsngl 33081* Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})

Theorembj-snglc 33082 Characterization of the elements of 𝐴 in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝐵 ↔ {𝐴} ∈ sngl 𝐵)

Theorembj-snglss 33083 The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 ⊆ 𝒫 𝐴

Theorembj-0nelsngl 33084 The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 7605). (Contributed by BJ, 6-Oct-2018.)
∅ ∉ sngl 𝐴

Theorembj-snglinv 33085* Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}

Theorembj-snglex 33086 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)

Syntaxbj-ctag 33087 Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
class tag 𝐴

Definitiondf-bj-tag 33088 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 ∪ {∅})

Theorembj-tageq 33089 Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

Theorembj-eltag 33090* Characterization of the elements of the tagging of a class. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ tag 𝐵 ↔ (∃𝑥𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))

Theorembj-0eltag 33091 The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
∅ ∈ tag 𝐴

Theorembj-tagn0 33092 The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
tag 𝐴 ≠ ∅

Theorembj-tagss 33093 The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
tag 𝐴 ⊆ 𝒫 𝐴

Theorembj-snglsstag 33094 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 ⊆ tag 𝐴

Theorembj-sngltagi 33095 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ sngl 𝐵𝐴 ∈ tag 𝐵)

Theorembj-sngltag 33096 The singletonization and the tagging of a set contain the same singletons. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))

Theorembj-tagci 33097 Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝐵 → {𝐴} ∈ tag 𝐵)

Theorembj-tagcg 33098 Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ∈ tag 𝐵))

Theorembj-taginv 33099* Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}

Theorembj-tagex 33100 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)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >