P. 343 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34201-34300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-rexcom4b 34201*	Remove from rexcom4b 3526 dependency on ax-ext 2795 and ax-13 2390 (and on df-or 844, df-cleq 2816, df-nfc 2965, df-v 3498). The hypothesis uses 𝑉 instead of V (see bj-isseti 34196 for the motivation). Use bj-rexcom4bv 34200 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ 𝐵 ∈ 𝑉    ⇒   ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)
Theorem	bj-ceqsalt0 34202	The FOL content of ceqsalt 3529. Lemma for bj-ceqsalt 34204 and bj-ceqsaltv 34205. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜃 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥𝜃) → (∀𝑥(𝜃 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalt1 34203	The FOL content of ceqsalt 3529. Lemma for bj-ceqsalt 34204 and bj-ceqsaltv 34205. TODO: consider removing if it does not add anything to bj-ceqsalt0 34202. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
⊢ (𝜃 → ∃𝑥𝜒)    ⇒   ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalt 34204*	Remove from ceqsalt 3529 dependency on ax-ext 2795 (and on df-cleq 2816 and df-v 3498). Note: this is not doable with ceqsralt 3530 (or ceqsralv 3535), which uses eleq1 2902, but the same dependence removal is possible for ceqsalg 3531, ceqsal 3533, ceqsalv 3534, cgsexg 3539, cgsex2g 3540, cgsex4g 3541, ceqsex 3542, ceqsexv 3543, ceqsex2 3545, ceqsex2v 3546, ceqsex3v 3547, ceqsex4v 3548, ceqsex6v 3549, ceqsex8v 3550, gencbvex 3551 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3552, gencbval 3553, vtoclgft 3555 (it uses Ⅎ, whose justification nfcjust 2964 does not use ax-ext 2795) and several other vtocl* theorems (see for instance bj-vtoclg1f 34236). See also bj-ceqsaltv 34205. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsaltv 34205*	Version of bj-ceqsalt 34204 with a disjoint variable condition on 𝑥, 𝑉, removing dependency on df-sb 2070 and df-clab 2802. Prefer its use over bj-ceqsalt 34204 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalg0 34206	The FOL content of ceqsalg 3531. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜒 → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalg 34207*	Remove from ceqsalg 3531 dependency on ax-ext 2795 (and on df-cleq 2816 and df-v 3498). See also bj-ceqsalgv 34209. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalgALT 34208*	Alternate proof of bj-ceqsalg 34207. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalgv 34209*	Version of bj-ceqsalg 34207 with a disjoint variable condition on 𝑥, 𝑉, removing dependency on df-sb 2070 and df-clab 2802. Prefer its use over bj-ceqsalg 34207 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsalgvALT 34210*	Alternate proof of bj-ceqsalgv 34209. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	bj-ceqsal 34211*	Remove from ceqsal 3533 dependency on ax-ext 2795 (and on df-cleq 2816, df-v 3498, df-clab 2802, df-sb 2070). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	bj-ceqsalv 34212*	Remove from ceqsalv 3534 dependency on ax-ext 2795 (and on df-cleq 2816, df-v 3498, df-clab 2802, df-sb 2070). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	bj-spcimdv 34213*	Remove from spcimdv 3594 dependency on ax-9 2124, ax-10 2145, ax-11 2161, ax-13 2390, ax-ext 2795, df-cleq 2816 (and df-nfc 2965, df-v 3498, df-or 844, df-tru 1540, df-nf 1785). For an even more economical version, see bj-spcimdvv 34214. (Contributed by BJ, 30-Nov-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-spcimdvv 34214*	Remove from spcimdv 3594 dependency on ax-7 2015, ax-8 2116, ax-10 2145, ax-11 2161, ax-12 2177 ax-13 2390, ax-ext 2795, df-cleq 2816, df-clab 2802 (and df-nfc 2965, df-v 3498, df-or 844, df-tru 1540, df-nf 1785) at the price of adding a disjoint variable condition on 𝑥, 𝐵 (but in usages, 𝑥 is typically a dummy, hence fresh, variable). For the version without this disjoint variable condition, see bj-spcimdv 34213. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
20.15.5.3  Characterization among sets versus among classes
Theorem	elelb 34215	Equivalence between two common ways to characterize elements of a class 𝐵: the LHS says that sets are elements of 𝐵 if and only if they satisfy 𝜑 while the RHS says that classes are elements of 𝐵 if and only if they are sets and satisfy 𝜑. Therefore, the LHS is a characterization among sets while the RHS is a characterization among classes. Note that the LHS is often formulated using a class variable instead of the universe V while this is not possible for the RHS (apart from using 𝐵 itself, which would not be very useful). (Contributed by BJ, 26-Feb-2023.)
⊢ ((𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜑)) ↔ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜑)))
Theorem	bj-pwvrelb 34216	Characterization of the elements of the powerclass of the cartesian square of the universal class: they are exactly the sets which are binary relations. (Contributed by BJ, 16-Dec-2023.)
⊢ (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴))
20.15.5.4  The nonfreeness quantifier for classes In this section, we prove the symmetry of the nonfreeness quantifier for classes.
Theorem	bj-nfcsym 34217	The nonfreeness quantifier for classes defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 5278 with additional axioms; see also nfcv 2979). This could be proved from aecom 2449 and nfcvb 5279 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 2829 instead of equcomd 2026; removing dependency on ax-ext 2795 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2999, eleq2d 2900 (using elequ2 2129), nfcvf 3009, dvelimc 3008, dvelimdc 3007, nfcvf2 3010. (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)
20.15.5.5  Proposal for the definitions of class membership and class equality Note: now that the proposals have been adopted as df-cleq 2816 and df-clel 2895, and that ax9ALT 2819 is in the main section, we only keep bj-ax9 34218 here as a weaker version of ax9ALT 2819 proved without ax-8 2116.
Theorem	bj-ax9 34218*	Proof of ax-9 2124 from Tarski's FOL=, sp 2182, dfcleq 2817 and ax-ext 2795 (with two extra disjoint variable conditions on 𝑥, 𝑧 and 𝑦, 𝑧). See ax9ALT 2819 for a more general version, proved using also ax-8 2116. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
20.15.5.6  Lemmas for class substitution Some useful theorems for dealing with substitutions: sbbi 2317, sbcbig 3825, sbcel1g 4367, sbcel2 4369, sbcel12 4362, sbceqg 4363, csbvarg 4385.
Theorem	bj-sbeqALT 34219*	Substitution in an equality (use the more general version bj-sbeq 34220 instead, without disjoint variable condition). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)
Theorem	bj-sbeq 34220	Distribute proper substitution through an equality relation. (See sbceqg 4363). (Contributed by BJ, 6-Oct-2018.)
⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)
Theorem	bj-sbceqgALT 34221	Distribute proper substitution through an equality relation. Alternate proof of sbceqg 4363. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 4363, but the Metamath program "MM-PA> MINIMIZE_WITH * / EXCEPT sbceqg" command is ok. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
Theorem	bj-csbsnlem 34222*	Lemma for bj-csbsn 34223 (in this lemma, 𝑥 cannot occur in 𝐴). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}
Theorem	bj-csbsn 34223	Substitution in a singleton. (Contributed by BJ, 6-Oct-2018.)
⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}
Theorem	bj-sbel1 34224*	Version of sbcel1g 4367 when substituting a set. (Note: one could have a corresponding version of sbcel12 4362 when substituting a set, but the point here is that the antecedent of sbcel1g 4367 is not needed when substituting a set.) (Contributed by BJ, 6-Oct-2018.)
⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵)
Theorem	bj-abv 34225	The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V)
Theorem	bj-ab0 34226	The class of sets verifying a falsity is the empty set (closed form of abf 4358). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
Theorem	bj-abf 34227	Shorter proof of abf 4358 (which should be kept as abfALT). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ {𝑥 ∣ 𝜑} = ∅
Theorem	bj-csbprc 34228	More direct proof of csbprc 4360 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
20.15.5.7  Removing some axiom requirements and disjoint variable conditions
Theorem	bj-exlimvmpi 34229*	A Fol lemma (exlimiv 1931 followed by mpi 20). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)
⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpi 34230	Lemma for bj-vtoclg1f1 34235 (an instance of this lemma is a version of bj-vtoclg1f1 34235 where 𝑥 and 𝑦 are identified). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpbi 34231	Lemma for theorems of the vtoclg 3569 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-exlimmpbir 34232	Lemma for theorems of the vtoclg 3569 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ (∃𝑥𝜒 → 𝜑)
Theorem	bj-vtoclf 34233*	Remove dependency on ax-ext 2795, df-clab 2802 and df-cleq 2816 (and df-sb 2070 and df-v 3498) from vtoclf 3560. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ 𝑉    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	bj-vtocl 34234*	Remove dependency on ax-ext 2795, df-clab 2802 and df-cleq 2816 (and df-sb 2070 and df-v 3498) from vtocl 3561. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝑉    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	bj-vtoclg1f1 34235*	The FOL content of vtoclg1f 3568 (hence not using ax-ext 2795, df-cleq 2816, df-nfc 2965, df-v 3498). Note the weakened "major" hypothesis and the disjoint variable condition between 𝑥 and 𝐴 (needed since the nonfreeness quantifier for classes is not available without ax-ext 2795; as a byproduct, this dispenses with ax-11 2161 and ax-13 2390). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (∃𝑦 𝑦 = 𝐴 → 𝜓)
Theorem	bj-vtoclg1f 34236*	Reprove vtoclg1f 3568 from bj-vtoclg1f1 34235. This removes dependency on ax-ext 2795, df-cleq 2816 and df-v 3498. Use bj-vtoclg1fv 34237 instead 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-vtoclg1fv 34237*	Version of bj-vtoclg1f 34236 with a disjoint variable condition on 𝑥, 𝑉. This removes dependency on df-sb 2070 and df-clab 2802. Prefer its use over bj-vtoclg1f 34236 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 34238*	A version of vtoclg 3569 with an additional disjoint variable condition (which is removable if we allow use of df-clab 2802, see bj-vtoclg1f 34236), which requires fewer axioms (i.e., removes dependency on ax-6 1970, ax-7 2015, ax-9 2124, ax-12 2177, ax-ext 2795, df-clab 2802, df-cleq 2816, df-v 3498). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)
Theorem	bj-rabbida2 34239	Version of rabbidva2 3478 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	bj-rabeqd 34240	Deduction form of rabeq 3485. Note that contrary to rabeq 3485 it has no disjoint variable condition. (Contributed by BJ, 27-Apr-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	bj-rabeqbid 34241	Version of rabeqbidv 3487 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-rabeqbida 34242	Version of rabeqbidva 3488 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 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	bj-seex 34243*	Version of seex 5520 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 34244*	Version of df-nfc 2965 with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 2-May-2019.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	bj-zfauscl 34245*	General version of zfauscl 5207. Remark: the comment in zfauscl 5207 is misleading: the essential use of ax-ext 2795 is the one via eleq2 2903 and not the one via vtocl 3561, since the latter can be proved without ax-ext 2795 (see bj-vtoclg 34238). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
20.15.5.8  Class abstractions A few additional theorems on class abstractions and restricted class abstractions.
Theorem	bj-unrab 34246*	Generalization of unrab 4276. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜓}) ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)}
Theorem	bj-inrab 34247	Generalization of inrab 4277. (Contributed by BJ, 21-Apr-2019.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐵 ∣ 𝜓}) = {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ (𝜑 ∧ 𝜓)}
Theorem	bj-inrab2 34248	Shorter proof of inrab 4277. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)}
Theorem	bj-inrab3 34249*	Generalization of dfrab3ss 4283, 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 34250*	Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
Theorem	bj-rabtrALT 34251*	Alternate proof of bj-rabtr 34250. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
Theorem	bj-rabtrAUTO 34252*	Proof of bj-rabtr 34250 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
20.15.5.9  Restricted nonfreeness In this subsection, we define restricted nonfreeness (or relative nonfreeness).
Syntax	wrnf 34253	Syntax for restricted nonfreeness.
wff Ⅎ𝑥 ∈ 𝐴𝜑
Definition	df-bj-rnf 34254	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))
20.15.5.10  Russell's paradox A few results around Russell's paradox. For clarity, we prove separately its FOL part (bj-ru0 34255) and then two versions (bj-ru1 34256 and bj-ru 34257). Special attention is put on minimizing axiom depencencies.
Theorem	bj-ru0 34255*	The FOL part of Russell's paradox ru 3773 (see also bj-ru1 34256, bj-ru 34257). Use of elequ1 2121, bj-elequ12 34014 (instead of eleq1 2902, eleq12d 2909 as in ru 3773) permits to remove dependency on ax-10 2145, ax-11 2161, ax-12 2177, ax-ext 2795, df-sb 2070, df-clab 2802, df-cleq 2816, df-clel 2895. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥)
Theorem	bj-ru1 34256*	A version of Russell's paradox ru 3773 (see also bj-ru 34257). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
Theorem	bj-ru 34257	Remove dependency on ax-13 2390 (and df-v 3498) from Russell's paradox ru 3773 expressed with primitive symbols and with a class variable 𝑉. Note the more economical use of bj-elissetv 34193 instead of isset 3508 to avoid use of df-v 3498. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉
20.15.5.11  Curry's paradox in set theory
Theorem	currysetlem 34258*	Lemma for currysetlem 34258, 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 34259*	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 34263. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉
Theorem	currysetlem1 34260*	Lemma for currysetALT 34263. (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 34261*	Lemma for currysetALT 34263. (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 34262*	Lemma for currysetALT 34263. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}    ⇒   ⊢ ¬ 𝑋 ∈ 𝑉
Theorem	currysetALT 34263*	Alternate proof of curryset 34259, 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
20.15.5.12  Some disjointness results A few utility theorems on disjointness of classes.
Theorem	bj-n0i 34264*	Inference associated with n0 4312. Shortens 2ndcdisj 22066 (2888>2878), notzfaus 5264 (264>253). (Contributed by BJ, 22-Apr-2019.)
⊢ 𝐴 ≠ ∅    ⇒   ⊢ ∃𝑥 𝑥 ∈ 𝐴
Theorem	bj-disjcsn 34265	A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 32009 and does not depend on df-ne 3019. (Contributed by BJ, 4-Apr-2019.)
⊢ (𝐴 ∩ {𝐴}) = ∅
Theorem	bj-disjsn01 34266	Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 34265 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)
⊢ ({∅} ∩ {1o}) = ∅
Theorem	bj-0nel1 34267	The empty set does not belong to {1o}. (Contributed by BJ, 6-Apr-2019.)
⊢ ∅ ∉ {1o}
Theorem	bj-1nel0 34268	1o does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.)
⊢ 1o ∉ {∅}
20.15.5.13  Complements on direct products A few utility theorems on direct products.
Theorem	bj-xpimasn 34269	The image of a singleton, general case. [Change and relabel xpimasn 6044 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
Theorem	bj-xpima1sn 34270	The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 6044 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Theorem	bj-xpima1snALT 34271	Alternate proof of bj-xpima1sn 34270. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Theorem	bj-xpima2sn 34272	The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6044] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Theorem	bj-xpnzex 34273	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 7627 (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 34274	Curried (exported) form of xpexg 7475. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × 𝐵) ∈ V))
Theorem	bj-xpnzexb 34275	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 34276*	Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)})
20.15.5.14  "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.
Theorem	bj-snsetex 34277*	The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5192. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
Theorem	bj-clex 34278*	Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐵)} ∈ V)
Syntax	bj-csngl 34279	Syntax for singletonization. (Contributed by BJ, 6-Oct-2018.)
class sngl 𝐴
Definition	df-bj-sngl 34280*	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 34281	Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
Theorem	bj-elsngl 34282*	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 34283	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 34284	The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 ⊆ 𝒫 𝐴
Theorem	bj-0nelsngl 34285	The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 8104). (Contributed by BJ, 6-Oct-2018.)
⊢ ∅ ∉ sngl 𝐴
Theorem	bj-snglinv 34286*	Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
Theorem	bj-snglex 34287	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 34288	Syntax for the tagged copy of a class. (Contributed by BJ, 6-Oct-2018.)
class tag 𝐴
Definition	df-bj-tag 34289	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 34290	Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)
Theorem	bj-eltag 34291*	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 34292	The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
⊢ ∅ ∈ tag 𝐴
Theorem	bj-tagn0 34293	The tagging of a class is nonempty. (Contributed by BJ, 6-Apr-2019.)
⊢ tag 𝐴 ≠ ∅
Theorem	bj-tagss 34294	The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
⊢ tag 𝐴 ⊆ 𝒫 𝐴
Theorem	bj-snglsstag 34295	The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ sngl 𝐴 ⊆ tag 𝐴
Theorem	bj-sngltagi 34296	The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ sngl 𝐵 → 𝐴 ∈ tag 𝐵)
Theorem	bj-sngltag 34297	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 34298	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 34299	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 34300*	Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}