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Theorem List for Metamath Proof Explorer - 31901-32000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-rexcom4b 31901* Remove from rexcom4b 3104 dependency on ax-ext 2494 and ax-13 2137 (and on df-or 383, df-cleq 2507, df-nfc 2644, df-v 3079). The hypothesis uses 𝑉 instead of V (see bj-isseti 31893 for the motivation). Use bj-rexcom4bv 31900 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
𝐵𝑉       (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)

Theorembj-ceqsalt0 31902 The FOL content of ceqsalt 3105. Lemma for bj-ceqsalt 31904 and bj-ceqsaltv 31905. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜃 → (𝜑𝜓)) ∧ ∃𝑥𝜃) → (∀𝑥(𝜃𝜑) ↔ 𝜓))

Theorembj-ceqsalt1 31903 The FOL content of ceqsalt 3105. Lemma for bj-ceqsalt 31904 and bj-ceqsaltv 31905. (TODO: consider removing if it does not add anything to bj-ceqsalt0 31902.) (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
(𝜃 → ∃𝑥𝜒)       ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) ↔ 𝜓))

Theorembj-ceqsalt 31904* Remove from ceqsalt 3105 dependency on ax-ext 2494 (and on df-cleq 2507 and df-v 3079). Note: this is not doable with ceqsralt 3106 (or ceqsralv 3111), which uses eleq1 2580, but the same dependence removal is possible for ceqsalg 3107, ceqsal 3109, ceqsalv 3110, cgsexg 3115, cgsex2g 3116, cgsex4g 3117, ceqsex 3118, ceqsexv 3119, ceqsex2 3121, ceqsex2v 3122, ceqsex3v 3123, ceqsex4v 3124, ceqsex6v 3125, ceqsex8v 3126, gencbvex 3127 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3128, gencbval 3129, vtoclgft 3131 (it uses , whose justification nfcjust 2643 is actually provable without ax-ext 2494, as bj-nfcjust 31879 shows) and several other vtocl* theorems (see for instance bj-vtoclg1f 31938). See also bj-ceqsaltv 31905. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theorembj-ceqsaltv 31905* Version of bj-ceqsalt 31904 with a dv condition on 𝑥, 𝑉, removing dependency on df-sb 1831 and df-clab 2501. Prefer its use over bj-ceqsalt 31904 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 31906 The FOL content of ceqsalg 3107. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒 → (𝜑𝜓))       (∃𝑥𝜒 → (∀𝑥(𝜒𝜑) ↔ 𝜓))

Theorembj-ceqsalg 31907* Remove from ceqsalg 3107 dependency on ax-ext 2494 (and on df-cleq 2507 and df-v 3079). See also bj-ceqsalgv 31909. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theorembj-ceqsalgALT 31908* Alternate proof of bj-ceqsalg 31907. (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 31909* Version of bj-ceqsalg 31907 with a dv condition on 𝑥, 𝑉, removing dependency on df-sb 1831 and df-clab 2501. Prefer its use over bj-ceqsalg 31907 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 31910* Alternate proof of bj-ceqsalgv 31909. (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 31911* Remove from ceqsal 3109 dependency on ax-ext 2494 (and on df-cleq 2507, df-v 3079, df-clab 2501, df-sb 1831). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)

Theorembj-ceqsalv 31912* Remove from ceqsalv 3110 dependency on ax-ext 2494 (and on df-cleq 2507, df-v 3079, df-clab 2501, df-sb 1831). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)

Theorembj-spcimdv 31913* Remove from spcimdv 3167 dependency on ax-10 1966, ax-11 1971, ax-13 2137, ax-ext 2494, df-cleq 2507 (and df-nfc 2644, df-v 3079, df-tru 1477, df-nf 1699). (Contributed by BJ, 30-Nov-2020.) (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 31914 The class-form not-free predicate defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 4722 with additional axioms; see also nfcv 2655). This could be proved from aecom 2203 and nfcvb 4723 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 2520 instead of equcomd 1896; removing dependency on ax-ext 2494 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2672, eleq2d 2577 (using elequ2 1952), nfcvf 2678, dvelimc 2677, dvelimdc 2676, nfcvf2 2679. (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 31915 and bj-ax9 31918) that the current forms of the definitions of class membership (df-clel 2510) and class equality (df-cleq 2507) are too powerful, and we propose alternate definitions (bj-df-clel 31916 and bj-df-cleq 31920) which also have the advantage of making it clear that these definitions are conservative.

Theorembj-ax8 31915 Proof of ax-8 1940 from df-clel 2510 (and FOL). This shows that df-clel 2510 is "too powerful". A possible definition is given by bj-df-clel 31916. (Contributed by BJ, 27-Jun-2019.) Also a direct consequence of bj-eleq1w 31875, which has essentially the same proof. (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theorembj-df-clel 31916* Candidate definition for df-clel 2510 (the need for it is exposed in bj-ax8 31915). 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 31917, which should be used instead. The proof is irrelevant since this is a proposal for an axiom.

Note: the current definition df-clel 2510 already mentions cleljust 1946 as a justification; here, we merely propose to put it as a hypothesis to make things clearer. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theorembj-dfclel 31917* Characterization of the elements of a class. Note: cleljust 1946 could be relabeled as clelhyp. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)
(𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))

Theorembj-ax9 31918* Proof of ax-9 1947 from ax-ext 2494 and df-cleq 2507 (and FOL) (with two extra dv conditions on 𝑥, 𝑦 and 𝑥, 𝑧). This shows that df-cleq 2507 is "too powerful". A possible definition is given by bj-df-cleq 31920. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theorembj-cleqhyp 31919* The hypothesis of bj-df-cleq 31920. Note that the hypothesis of bj-df-cleq 31920 actually has an additional dv condition on 𝑥, 𝑦 and therefore is provable by simply using ax-ext 2494 in place of axext3 2496 in the current proof. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))

Theorembj-df-cleq 31920* Candidate definition for df-cleq 2507 (the need for it is exposed in bj-ax9 31918). 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 31921, 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.) (New usage is discouraged.)

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

Theorembj-dfcleq 31921* Proof of class extensionality from the axiom of set extensionality (ax-ext 2494) and the definition of class equality (bj-df-cleq 31920). (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 2293, sbcbig 3351, sbcel1g 3842, sbcel2 3844, sbcel12 3838, sbceqg 3839, csbvarg 3858.

Theorembj-sbeqALT 31922* Substitution in an equality (use the more genereal version bj-sbeq 31923 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 31923 Distribute proper substitution through an equality relation. (See sbceqg 3839). (Contributed by BJ, 6-Oct-2018.)
([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)

Theorembj-sbceqgALT 31924 Distribute proper substitution through an equality relation. Alternate proof of sbceqg 3839. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 3839, but "minimize */except sbceqg" is ok. (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

Theorembj-csbsnlem 31925* Lemma for bj-csbsn 31926 (in this lemma, 𝑥 cannot occur in 𝐴). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
𝐴 / 𝑥{𝑥} = {𝐴}

Theorembj-csbsn 31926 Substitution in a singleton. (Contributed by BJ, 6-Oct-2018.)
𝐴 / 𝑥{𝑥} = {𝐴}

Theorembj-sbel1 31927* Version of sbcel1g 3842 when substituting a set. (Note: one could have a corresponding version of sbcel12 3838 when substituting a set, but the point here is that the antecedent of sbcel1g 3842 is not needed when substituting a set.) (Contributed by BJ, 6-Oct-2018.)
([𝑦 / 𝑥]𝐴𝐵𝑦 / 𝑥𝐴𝐵)

Theorembj-abv 31928 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 31929 The class of sets verifying a falsity is the empty set (closed form of abf 3833). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
(∀𝑥 ¬ 𝜑 → {𝑥𝜑} = ∅)

Theorembj-abf 31930 Shorter proof of abf 3833 (which should be kept as abfALT). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
¬ 𝜑       {𝑥𝜑} = ∅

Theorembj-csbprc 31931 More direct proof of csbprc 3835 (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 31932 Lemma for bj-vtoclg1f1 31937 (an instance of this lemma is a version of bj-vtoclg1f1 31937 where 𝑥 and 𝑦 are identified). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒 → (𝜑𝜓))    &   𝜑       (∃𝑥𝜒𝜓)

Theorembj-exlimmpbi 31933 Lemma for theorems of the vtoclg 3143 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒 → (𝜑𝜓))    &   𝜑       (∃𝑥𝜒𝜓)

Theorembj-exlimmpbir 31934 Lemma for theorems of the vtoclg 3143 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜒 → (𝜑𝜓))    &   𝜓       (∃𝑥𝜒𝜑)

Theorembj-vtoclf 31935* Remove dependency on ax-ext 2494, df-clab 2501 and df-cleq 2507 (and df-sb 1831 and df-v 3079) from vtoclf 3135. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   𝐴𝑉    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓

Theorembj-vtocl 31936* Remove dependency on ax-ext 2494, df-clab 2501 and df-cleq 2507 (and df-sb 1831 and df-v 3079) from vtocl 3136. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
𝐴𝑉    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓

Theorembj-vtoclg1f1 31937* The FOL content of vtoclg1f 3142 (hence not using ax-ext 2494, df-cleq 2507, df-nfc 2644, df-v 3079). 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 2494; as a byproduct, this dispenses with ax-11 1971 and ax-13 2137). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (∃𝑦 𝑦 = 𝐴𝜓)

Theorembj-vtoclg1f 31938* Reprove vtoclg1f 3142 from bj-vtoclg1f1 31937. This removes dependency on ax-ext 2494, df-cleq 2507 and df-v 3079. Use bj-vtoclg1fv 31939 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 31939* Version of bj-vtoclg1f 31938 with a dv condition on 𝑥, 𝑉. This removes dependency on df-sb 1831 and df-clab 2501. Prefer its use over bj-vtoclg1f 31938 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 31940 Version of rabbidva2 3066 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theorembj-rabbida 31941 Version of rabbidva 3067 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theorembj-rabbid 31942 Version of rabbidv 3068 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theorembj-rabeqd 31943 Deduction form of rabeq 3070. Note that contrary to rabeq 3070 it has no dv condition. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})

Theorembj-rabeqbid 31944 Version of rabeqbidv 3072 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 31945 Version of rabeqbidva 3073 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 31946* Version of seex 4895 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 31947* Version of df-nfc 2644 with a dv condition replaced with a non-freeness hypothesis. (Contributed by BJ, 2-May-2019.)
𝑦𝐴       (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)

Theorembj-axsep2 31948* Remove dependency on ax-8 1940, ax-10 1966, ax-12 1983, ax-13 2137, ax-ext 2494, df-cleq 2507 and df-clel 2510 from axsep2 4608 while shortening its proof (note that axsep2 4608 does require ax-8 1940 and ax-9 1947 since it requires df-clel 2510 and df-cleq 2507--- see bj-df-clel 31916 and bj-df-cleq 31920). Remark: the comment in zfauscl 4609 is misleading: the essential use of ax-ext 2494 is the one via eleq2 2581 and not the one via vtocl 3136, since the latter can be proved without ax-ext 2494 (see bj-vtocl 31936). (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 31949* Generalization of unrab 3760. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)
({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜓}) ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}

Theorembj-inrab 31950 Generalization of inrab 3761. (Contributed by BJ, 21-Apr-2019.)
({𝑥𝐴𝜑} ∩ {𝑥𝐵𝜓}) = {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}

Theorembj-inrab2 31951 Shorter proof of inrab 3761. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Theorembj-inrab3 31952* Generalization of dfrab3ss 3767, which it may shorten. (Contributed by BJ, 21-Apr-2019.) (Revised by OpenAI, 7-Jul-2020.)
(𝐴 ∩ {𝑥𝐵𝜑}) = ({𝑥𝐴𝜑} ∩ 𝐵)

Theorembj-rabtr 31953* Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.)
{𝑥𝐴 ∣ ⊤} = 𝐴

Theorembj-rabtrALT 31954* Alternate proof of bj-rabtr 31953. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝐴 ∣ ⊤} = 𝐴

Theorembj-rabtrALTALT 31955* Alternate proof of bj-rabtr 31953. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝐴 ∣ ⊤} = 𝐴

Theorembj-rabtrAUTO 31956* Proof of bj-rabtr 31953 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 31957 Syntax for restricted non-freeness.
wff 𝑥𝐴𝜑

Definitiondf-bj-rnf 31958 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 31959) and then two versions (bj-ru1 31960 and bj-ru 31961). Special attention is put on minimizing axiom depencencies.

Theorembj-ru0 31959* The FOL part of Russell's paradox ru 3305 (see also bj-ru1 31960, bj-ru 31961). Use of elequ1 1945, bj-elequ12 31693, bj-spvv 31748 (instead of eleq1 2580, eleq12d 2586, spv 2151 as in ru 3305) permits to remove dependency on ax-10 1966, ax-11 1971, ax-12 1983, ax-13 2137, ax-ext 2494, df-sb 1831, df-clab 2501, df-cleq 2507, df-clel 2510. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥)

Theorembj-ru1 31960* A version of Russell's paradox ru 3305 (see also bj-ru 31961). Note the more economical use of bj-abeq2 31806 instead of abeq2 2623 to avoid dependency on ax-13 2137. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}

Theorembj-ru 31961 Remove dependency on ax-13 2137 (and df-v 3079) from Russell's paradox ru 3305 expressed with primitive symbols and with a class variable 𝑉 (note that axsep2 4608 does require ax-8 1940 and ax-9 1947 since it requires df-clel 2510 and df-cleq 2507--- see bj-df-clel 31916 and bj-df-cleq 31920). Note the more economical use of bj-elissetv 31890 instead of isset 3084 to avoid use of df-v 3079. (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 31962* Inference associated with n0 3793. (Minimizes three statements by a total of 29 bytes.) (Contributed by BJ, 22-Apr-2019.)
𝐴 ≠ ∅       𝑥 𝑥𝐴

Theorembj-nel0 31963* From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. [Could shorten 0xp 5016?] (Contributed by BJ, 6-Oct-2018.)
¬ 𝑥𝐴       𝐴 = ∅

Theorembj-disjcsn 31964 A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 29908. (Contributed by BJ, 4-Apr-2019.)
(𝐴 ∩ {𝐴}) = ∅

Theorembj-disjsn01 31965 Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 31964 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)
({∅} ∩ {1𝑜}) = ∅

Theorembj-1ex 31966 1𝑜 is a set. (Contributed by BJ, 6-Apr-2019.)
1𝑜 ∈ V

Theorembj-2ex 31967 2𝑜 is a set. (Contributed by BJ, 6-Apr-2019.)
2𝑜 ∈ V

Theorembj-0nel1 31968 The empty set does not belong to {1𝑜}. (Contributed by BJ, 6-Apr-2019.)
∅ ∉ {1𝑜}

Theorembj-1nel0 31969 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 31970 The image of a singleton, general case. [Change and relabel xpimasn 5388 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)

Theorembj-xpima1sn 31971 The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 5388 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Theorembj-xpima1snALT 31972 Alternate proof of bj-xpima1sn 31971. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Theorembj-xpima2sn 31973 The image of a singleton by a direct product, nonempty case. [To replace xpimasn 5388] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Theorembj-xpnzex 31974 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 exported form (curried form) of xpexcnv 6881 (up to commutation in the product). (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
(𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝑉𝐵 ∈ V))

Theorembj-xpexg2 31975 Exported form (curried form) of xpexg 6739. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → (𝐵𝑊 → (𝐴 × 𝐵) ∈ V))

Theorembj-xpnzexb 31976 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 31977* 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 31978* 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 31979* The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 4597. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)

Theorembj-clex 31980* Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐵)} ∈ V)

Syntaxbj-csngl 31981 Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
class sngl 𝐴

Definitiondf-bj-sngl 31982* 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 31983 Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)

Theorembj-elsngl 31984* Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})

Theorembj-snglc 31985 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 31986 The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 ⊆ 𝒫 𝐴

Theorembj-0nelsngl 31987 The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 7327). (Contributed by BJ, 6-Oct-2018.)
∅ ∉ sngl 𝐴

Theorembj-snglinv 31988* Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}

Theorembj-snglex 31989 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 31990 Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
class tag 𝐴

Definitiondf-bj-tag 31991 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 31992 Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

Theorembj-eltag 31993* 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 31994 The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
∅ ∈ tag 𝐴

Theorembj-tagn0 31995 The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
tag 𝐴 ≠ ∅

Theorembj-tagss 31996 The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
tag 𝐴 ⊆ 𝒫 𝐴

Theorembj-snglsstag 31997 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
sngl 𝐴 ⊆ tag 𝐴

Theorembj-sngltagi 31998 The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
(𝐴 ∈ sngl 𝐵𝐴 ∈ tag 𝐵)

Theorembj-sngltag 31999 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 32000 Characterization of the elements of 𝐵 in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝐵 → {𝐴} ∈ tag 𝐵)

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