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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))
 
20.14.5.9  Russell's paradox

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)
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