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Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembdceqi 10901 A class equal to a bounded one is bounded. Note the use of ax-ext 2065. See also bdceqir 10902. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   𝐴 = 𝐵       BOUNDED 𝐵
 
Theorembdceqir 10902 A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 10901) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 10883). (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   𝐵 = 𝐴       BOUNDED 𝐵
 
Theorembdel 10903* The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
(BOUNDED 𝐴BOUNDED 𝑥𝐴)
 
Theorembdeli 10904* Inference associated with bdel 10903. Its converse is bdelir 10905. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑥𝐴
 
Theorembdelir 10905* Inference associated with df-bdc 10899. Its converse is bdeli 10904. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝑥𝐴       BOUNDED 𝐴
 
Theorembdcv 10906 A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝑥
 
Theorembdcab 10907 A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
BOUNDED 𝜑       BOUNDED {𝑥𝜑}
 
Theorembdph 10908 A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
BOUNDED {𝑥𝜑}       BOUNDED 𝜑
 
Theorembds 10909* Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 10880; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 10880. (Contributed by BJ, 19-Nov-2019.)
BOUNDED 𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       BOUNDED 𝜓
 
Theorembdcrab 10910* A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   BOUNDED 𝜑       BOUNDED {𝑥𝐴𝜑}
 
Theorembdne 10911 Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝑥𝑦
 
Theorembdnel 10912* Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑥𝐴
 
Theorembdreu 10913* Boundedness of existential uniqueness.

Remark regarding restricted quantifiers: the formula 𝑥𝐴𝜑 need not be bounded even if 𝐴 and 𝜑 are. Indeed, V is bounded by bdcvv 10915, and (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 10882, if 𝑥 ∈ V𝜑 were bounded when 𝜑 is bounded, then 𝑥𝜑 would be bounded as well when 𝜑 is bounded, which is not the case. The same remark holds with ∃, ∃!, ∃*. (Contributed by BJ, 16-Oct-2019.)

BOUNDED 𝜑       BOUNDED ∃!𝑥𝑦 𝜑
 
Theorembdrmo 10914* Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝜑       BOUNDED ∃*𝑥𝑦 𝜑
 
Theorembdcvv 10915 The universal class is bounded. The formulation may sound strange, but recall that here, "bounded" means "Δ0". (Contributed by BJ, 3-Oct-2019.)
BOUNDED V
 
Theorembdsbc 10916 A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 10917. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝜑       BOUNDED [𝑦 / 𝑥]𝜑
 
TheorembdsbcALT 10917 Alternate proof of bdsbc 10916. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
BOUNDED 𝜑       BOUNDED [𝑦 / 𝑥]𝜑
 
Theorembdccsb 10918 A class resulting from proper substitution of a setvar for a setvar in a bounded class is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑦 / 𝑥𝐴
 
Theorembdcdif 10919 The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   BOUNDED 𝐵       BOUNDED (𝐴𝐵)
 
Theorembdcun 10920 The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   BOUNDED 𝐵       BOUNDED (𝐴𝐵)
 
Theorembdcin 10921 The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   BOUNDED 𝐵       BOUNDED (𝐴𝐵)
 
Theorembdss 10922 The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑥𝐴
 
Theorembdcnul 10923 The empty class is bounded. See also bdcnulALT 10924. (Contributed by BJ, 3-Oct-2019.)
BOUNDED
 
TheorembdcnulALT 10924 Alternate proof of bdcnul 10923. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 10902, or use the corresponding characterizations of its elements followed by bdelir 10905. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
BOUNDED
 
Theorembdeq0 10925 Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)
BOUNDED 𝑥 = ∅
 
Theorembj-bd0el 10926 Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.)
BOUNDED ∅ ∈ 𝑥
 
Theorembdcpw 10927 The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝒫 𝐴
 
Theorembdcsn 10928 The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED {𝑥}
 
Theorembdcpr 10929 The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED {𝑥, 𝑦}
 
Theorembdctp 10930 The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED {𝑥, 𝑦, 𝑧}
 
Theorembdsnss 10931* Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝐴       BOUNDED {𝑥} ⊆ 𝐴
 
Theorembdvsn 10932* Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝑥 = {𝑦}
 
Theorembdop 10933 The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
BOUNDED𝑥, 𝑦
 
Theorembdcuni 10934 The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
BOUNDED 𝑥
 
Theorembdcint 10935 The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝑥
 
Theorembdciun 10936* The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑥𝑦 𝐴
 
Theorembdciin 10937* The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑥𝑦 𝐴
 
Theorembdcsuc 10938 The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED suc 𝑥
 
Theorembdeqsuc 10939* Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
BOUNDED 𝑥 = suc 𝑦
 
Theorembj-bdsucel 10940 Boundedness of the formula "the successor of the setvar 𝑥 belongs to the setvar 𝑦". (Contributed by BJ, 30-Nov-2019.)
BOUNDED suc 𝑥𝑦
 
Theorembdcriota 10941* A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
BOUNDED 𝜑    &   ∃!𝑥𝑦 𝜑       BOUNDED (𝑥𝑦 𝜑)
 
6.2.6  CZF: Bounded separation

In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.

 
Axiomax-bdsep 10942* Axiom scheme of bounded (or restricted, or Δ0) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 3916. (Contributed by BJ, 5-Oct-2019.)
BOUNDED 𝜑       𝑎𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
Theorembdsep1 10943* Version of ax-bdsep 10942 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
Theorembdsep2 10944* Version of ax-bdsep 10942 with one DV condition removed and without initial universal quantifier. Use bdsep1 10943 when sufficient. (Contributed by BJ, 5-Oct-2019.)
BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
Theorembdsepnft 10945* Closed form of bdsepnf 10946. Version of ax-bdsep 10942 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 10943 when sufficient. (Contributed by BJ, 19-Oct-2019.)
BOUNDED 𝜑       (∀𝑥𝑏𝜑 → ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
 
Theorembdsepnf 10946* Version of ax-bdsep 10942 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 10947. Use bdsep1 10943 when sufficient. (Contributed by BJ, 5-Oct-2019.)
𝑏𝜑    &   BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
TheorembdsepnfALT 10947* Alternate proof of bdsepnf 10946, not using bdsepnft 10945. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏𝜑    &   BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
Theorembdzfauscl 10948* Closed form of the version of zfauscl 3918 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
BOUNDED 𝜑       (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
 
Theorembdbm1.3ii 10949* Bounded version of bm1.3ii 3919. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)
 
Theorembj-axemptylem 10950* Lemma for bj-axempty 10951 and bj-axempty2 10952. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3924 instead. (New usage is discouraged.)
𝑥𝑦(𝑦𝑥 → ⊥)
 
Theorembj-axempty 10951* Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a non-empty universe. See axnul 3923. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3924 instead. (New usage is discouraged.)
𝑥𝑦𝑥
 
Theorembj-axempty2 10952* Axiom of the empty set from bounded separation, alternate version to bj-axempty 10951. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3924 instead. (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥
 
Theorembj-nalset 10953* nalset 3928 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ ∃𝑥𝑦 𝑦𝑥
 
Theorembj-vprc 10954 vprc 3929 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ V ∈ V
 
Theorembj-nvel 10955 nvel 3930 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ V ∈ 𝐴
 
Theorembj-vnex 10956 vnex 3931 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ ∃𝑥 𝑥 = V
 
Theorembdinex1 10957 Bounded version of inex1 3932. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵    &   𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theorembdinex2 10958 Bounded version of inex2 3933. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵    &   𝐴 ∈ V       (𝐵𝐴) ∈ V
 
Theorembdinex1g 10959 Bounded version of inex1g 3934. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵       (𝐴𝑉 → (𝐴𝐵) ∈ V)
 
Theorembdssex 10960 Bounded version of ssex 3935. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)
 
Theorembdssexi 10961 Bounded version of ssexi 3936. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V
 
Theorembdssexg 10962 Bounded version of ssexg 3937. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theorembdssexd 10963 Bounded version of ssexd 3938. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)    &   BOUNDED 𝐴       (𝜑𝐴 ∈ V)
 
Theorembdrabexg 10964* Bounded version of rabexg 3941. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   BOUNDED 𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 
Theorembj-inex 10965 The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theorembj-intexr 10966 intexr 3945 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 ∈ V → 𝐴 ≠ ∅)
 
Theorembj-intnexr 10967 intnexr 3946 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 = V → ¬ 𝐴 ∈ V)
 
Theorembj-zfpair2 10968 Proof of zfpair2 3993 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
{𝑥, 𝑦} ∈ V
 
Theorembj-prexg 10969 Proof of prexg 3994 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
 
Theorembj-snexg 10970 snexg 3976 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝐴} ∈ V)
 
Theorembj-snex 10971 snex 3977 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       {𝐴} ∈ V
 
Theorembj-sels 10972* If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.)
(𝐴𝑉 → ∃𝑥 𝐴𝑥)
 
Theorembj-axun2 10973* axun2 4218 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
 
Theorembj-uniex2 10974* uniex2 4219 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦 𝑦 = 𝑥
 
Theorembj-uniex 10975 uniex 4220 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V        𝐴 ∈ V
 
Theorembj-uniexg 10976 uniexg 4221 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 𝐴 ∈ V)
 
Theorembj-unex 10977 unex 4222 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V
 
Theorembdunexb 10978 Bounded version of unexb 4223. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝐵       ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
 
Theorembj-unexg 10979 unexg 4224 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theorembj-sucexg 10980 sucexg 4270 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → suc 𝐴 ∈ V)
 
Theorembj-sucex 10981 sucex 4271 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       suc 𝐴 ∈ V
 
6.2.6.1  Delta_0-classical logic
 
Axiomax-bj-d0cl 10982 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
BOUNDED 𝜑       DECID 𝜑
 
Theorembj-notbi 10983 Equivalence property for negation. TODO: minimize all theorems using notbid 625 and notbii 627. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
 
Theorembj-notbii 10984 Inference associated with bj-notbi 10983. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
(𝜑𝜓)       𝜑 ↔ ¬ 𝜓)
 
Theorembj-notbid 10985 Deduction form of bj-notbi 10983. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
 
Theorembj-dcbi 10986 Equivalence property for DECID. TODO: solve conflict with dcbi 878; minimize dcbii 781 and dcbid 782 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
((𝜑𝜓) → (DECID 𝜑DECID 𝜓))
 
Theorembj-d0clsepcl 10987 Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
DECID 𝜑
 
6.2.6.2  Inductive classes and the class of natural numbers (finite ordinals)
 
Syntaxwind 10988 Syntax for inductive classes.
wff Ind 𝐴
 
Definitiondf-bj-ind 10989* Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
 
Theorembj-indsuc 10990 A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
 
Theorembj-indeq 10991 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
(𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
 
Theorembj-bdind 10992 Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
BOUNDED Ind 𝑥
 
Theorembj-indint 10993* The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Ind {𝑥𝐴 ∣ Ind 𝑥}
 
Theorembj-indind 10994* If 𝐴 is inductive and 𝐵 is "inductive in 𝐴", then (𝐴𝐵) is inductive. (Contributed by BJ, 25-Oct-2020.)
((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → Ind (𝐴𝐵))
 
Theorembj-dfom 10995 Alternate definition of ω, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)
ω = {𝑥 ∣ Ind 𝑥}
 
Theorembj-omind 10996 ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
Ind ω
 
Theorembj-omssind 10997 ω is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))
 
Theorembj-ssom 10998* A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)
 
Theorembj-om 10999* A set is equal to ω if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥))))
 
Theorembj-2inf 11000* Two formulations of the axiom of infinity (see ax-infvn 11003 and bj-omex 11004) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
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