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Theorem List for Intuitionistic Logic Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-stim 14501 A conjunction with a stable consequent is stable. See stabnot 833 for negation , bj-stan 14502 for conjunction , and bj-stal 14504 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
(STAB πœ“ β†’ STAB (πœ‘ β†’ πœ“))
 
Theorembj-stan 14502 The conjunction of two stable formulas is stable. See bj-stim 14501 for implication, stabnot 833 for negation, and bj-stal 14504 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
((STAB πœ‘ ∧ STAB πœ“) β†’ STAB (πœ‘ ∧ πœ“))
 
Theorembj-stand 14503 The conjunction of two stable formulas is stable. Deduction form of bj-stan 14502. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 14502 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(πœ‘ β†’ STAB πœ“)    &   (πœ‘ β†’ STAB πœ’)    β‡’   (πœ‘ β†’ STAB (πœ“ ∧ πœ’))
 
Theorembj-stal 14504 The universal quantification of a stable formula is stable. See bj-stim 14501 for implication, stabnot 833 for negation, and bj-stan 14502 for conjunction. (Contributed by BJ, 24-Nov-2023.)
(βˆ€π‘₯STAB πœ‘ β†’ STAB βˆ€π‘₯πœ‘)
 
Theorembj-pm2.18st 14505 Clavius law for stable formulas. See pm2.18dc 855. (Contributed by BJ, 4-Dec-2023.)
(STAB πœ‘ β†’ ((Β¬ πœ‘ β†’ πœ‘) β†’ πœ‘))
 
Theorembj-con1st 14506 Contraposition when the antecedent is a negated stable proposition. See con1dc 856. (Contributed by BJ, 11-Nov-2024.)
(STAB πœ‘ β†’ ((Β¬ πœ‘ β†’ πœ“) β†’ (Β¬ πœ“ β†’ πœ‘)))
 
12.2.1.2  Decidable formulas
 
Theorembj-trdc 14507 A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(πœ‘ β†’ DECID πœ‘)
 
Theorembj-dctru 14508 The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊀
 
Theorembj-fadc 14509 A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(Β¬ πœ‘ β†’ DECID πœ‘)
 
Theorembj-dcfal 14510 The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID βŠ₯
 
Theorembj-dcstab 14511 A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(DECID πœ‘ β†’ STAB πœ‘)
 
Theorembj-nnbidc 14512 If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 14499. (Contributed by BJ, 24-Nov-2023.)
(Β¬ Β¬ πœ‘ β†’ (DECID πœ‘ ↔ πœ‘))
 
Theorembj-nndcALT 14513 Alternate proof of nndc 851. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)
Β¬ Β¬ DECID πœ‘
 
Theorembj-dcdc 14514 Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)
(DECID DECID πœ‘ ↔ DECID πœ‘)
 
Theorembj-stdc 14515 Decidability of a proposition is stable if and only if that proposition is decidable. In particular, the assumption that every formula is stable implies that every formula is decidable, hence classical logic. (Contributed by BJ, 9-Oct-2019.)
(STAB DECID πœ‘ ↔ DECID πœ‘)
 
Theorembj-dcst 14516 Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)
(DECID STAB πœ‘ ↔ STAB πœ‘)
 
12.2.2  Predicate calculus
 
Theorembj-ex 14517* Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1598 and 19.9ht 1641 or 19.23ht 1497). (Proof modification is discouraged.)
(βˆƒπ‘₯πœ‘ β†’ πœ‘)
 
Theorembj-hbalt 14518 Closed form of hbal 1477 (copied from set.mm). (Contributed by BJ, 2-May-2019.)
(βˆ€π‘¦(πœ‘ β†’ βˆ€π‘₯πœ‘) β†’ (βˆ€π‘¦πœ‘ β†’ βˆ€π‘₯βˆ€π‘¦πœ‘))
 
Theorembj-nfalt 14519 Closed form of nfal 1576 (copied from set.mm). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
(βˆ€π‘₯β„²π‘¦πœ‘ β†’ β„²π‘¦βˆ€π‘₯πœ‘)
 
Theoremspimd 14520 Deduction form of spim 1738. (Contributed by BJ, 17-Oct-2019.)
(πœ‘ β†’ β„²π‘₯πœ’)    &   (πœ‘ β†’ βˆ€π‘₯(π‘₯ = 𝑦 β†’ (πœ“ β†’ πœ’)))    β‡’   (πœ‘ β†’ (βˆ€π‘₯πœ“ β†’ πœ’))
 
Theorem2spim 14521* Double substitution, as in spim 1738. (Contributed by BJ, 17-Oct-2019.)
β„²π‘₯πœ’    &   β„²π‘§πœ’    &   ((π‘₯ = 𝑦 ∧ 𝑧 = 𝑑) β†’ (πœ“ β†’ πœ’))    β‡’   (βˆ€π‘§βˆ€π‘₯πœ“ β†’ πœ’)
 
Theoremch2var 14522* Implicit substitution of 𝑦 for π‘₯ and 𝑑 for 𝑧 into a theorem. (Contributed by BJ, 17-Oct-2019.)
β„²π‘₯πœ“    &   β„²π‘§πœ“    &   ((π‘₯ = 𝑦 ∧ 𝑧 = 𝑑) β†’ (πœ‘ ↔ πœ“))    &   πœ‘    β‡’   πœ“
 
Theoremch2varv 14523* Version of ch2var 14522 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)
((π‘₯ = 𝑦 ∧ 𝑧 = 𝑑) β†’ (πœ‘ ↔ πœ“))    &   πœ‘    β‡’   πœ“
 
Theorembj-exlimmp 14524 Lemma for bj-vtoclgf 14531. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
β„²π‘₯πœ“    &   (πœ’ β†’ πœ‘)    β‡’   (βˆ€π‘₯(πœ’ β†’ (πœ‘ β†’ πœ“)) β†’ (βˆƒπ‘₯πœ’ β†’ πœ“))
 
Theorembj-exlimmpi 14525 Lemma for bj-vtoclgf 14531. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
β„²π‘₯πœ“    &   (πœ’ β†’ πœ‘)    &   (πœ’ β†’ (πœ‘ β†’ πœ“))    β‡’   (βˆƒπ‘₯πœ’ β†’ πœ“)
 
Theorembj-sbimedh 14526 A strengthening of sbiedh 1787 (same proof). (Contributed by BJ, 16-Dec-2019.)
(πœ‘ β†’ βˆ€π‘₯πœ‘)    &   (πœ‘ β†’ (πœ’ β†’ βˆ€π‘₯πœ’))    &   (πœ‘ β†’ (π‘₯ = 𝑦 β†’ (πœ“ β†’ πœ’)))    β‡’   (πœ‘ β†’ ([𝑦 / π‘₯]πœ“ β†’ πœ’))
 
Theorembj-sbimeh 14527 A strengthening of sbieh 1790 (same proof). (Contributed by BJ, 16-Dec-2019.)
(πœ“ β†’ βˆ€π‘₯πœ“)    &   (π‘₯ = 𝑦 β†’ (πœ‘ β†’ πœ“))    β‡’   ([𝑦 / π‘₯]πœ‘ β†’ πœ“)
 
Theorembj-sbime 14528 A strengthening of sbie 1791 (same proof). (Contributed by BJ, 16-Dec-2019.)
β„²π‘₯πœ“    &   (π‘₯ = 𝑦 β†’ (πœ‘ β†’ πœ“))    β‡’   ([𝑦 / π‘₯]πœ‘ β†’ πœ“)
 
12.2.3  Set theorey miscellaneous
 
Theorembj-el2oss1o 14529 Shorter proof of el2oss1o 6444 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 2o β†’ 𝐴 βŠ† 1o)
 
12.2.4  Extensionality

Various utility theorems using FOL and extensionality.

 
Theorembj-vtoclgft 14530 Weakening two hypotheses of vtoclgf 2796. (Contributed by BJ, 21-Nov-2019.)
β„²π‘₯𝐴    &   β„²π‘₯πœ“    &   (π‘₯ = 𝐴 β†’ πœ‘)    β‡’   (βˆ€π‘₯(π‘₯ = 𝐴 β†’ (πœ‘ β†’ πœ“)) β†’ (𝐴 ∈ 𝑉 β†’ πœ“))
 
Theorembj-vtoclgf 14531 Weakening two hypotheses of vtoclgf 2796. (Contributed by BJ, 21-Nov-2019.)
β„²π‘₯𝐴    &   β„²π‘₯πœ“    &   (π‘₯ = 𝐴 β†’ πœ‘)    &   (π‘₯ = 𝐴 β†’ (πœ‘ β†’ πœ“))    β‡’   (𝐴 ∈ 𝑉 β†’ πœ“)
 
Theoremelabgf0 14532 Lemma for elabgf 2880. (Contributed by BJ, 21-Nov-2019.)
(π‘₯ = 𝐴 β†’ (𝐴 ∈ {π‘₯ ∣ πœ‘} ↔ πœ‘))
 
Theoremelabgft1 14533 One implication of elabgf 2880, in closed form. (Contributed by BJ, 21-Nov-2019.)
β„²π‘₯𝐴    &   β„²π‘₯πœ“    β‡’   (βˆ€π‘₯(π‘₯ = 𝐴 β†’ (πœ‘ β†’ πœ“)) β†’ (𝐴 ∈ {π‘₯ ∣ πœ‘} β†’ πœ“))
 
Theoremelabgf1 14534 One implication of elabgf 2880. (Contributed by BJ, 21-Nov-2019.)
β„²π‘₯𝐴    &   β„²π‘₯πœ“    &   (π‘₯ = 𝐴 β†’ (πœ‘ β†’ πœ“))    β‡’   (𝐴 ∈ {π‘₯ ∣ πœ‘} β†’ πœ“)
 
Theoremelabgf2 14535 One implication of elabgf 2880. (Contributed by BJ, 21-Nov-2019.)
β„²π‘₯𝐴    &   β„²π‘₯πœ“    &   (π‘₯ = 𝐴 β†’ (πœ“ β†’ πœ‘))    β‡’   (𝐴 ∈ 𝐡 β†’ (πœ“ β†’ 𝐴 ∈ {π‘₯ ∣ πœ‘}))
 
Theoremelabf1 14536* One implication of elabf 2881. (Contributed by BJ, 21-Nov-2019.)
β„²π‘₯πœ“    &   (π‘₯ = 𝐴 β†’ (πœ‘ β†’ πœ“))    β‡’   (𝐴 ∈ {π‘₯ ∣ πœ‘} β†’ πœ“)
 
Theoremelabf2 14537* One implication of elabf 2881. (Contributed by BJ, 21-Nov-2019.)
β„²π‘₯πœ“    &   π΄ ∈ V    &   (π‘₯ = 𝐴 β†’ (πœ“ β†’ πœ‘))    β‡’   (πœ“ β†’ 𝐴 ∈ {π‘₯ ∣ πœ‘})
 
Theoremelab1 14538* One implication of elab 2882. (Contributed by BJ, 21-Nov-2019.)
(π‘₯ = 𝐴 β†’ (πœ‘ β†’ πœ“))    β‡’   (𝐴 ∈ {π‘₯ ∣ πœ‘} β†’ πœ“)
 
Theoremelab2a 14539* One implication of elab 2882. (Contributed by BJ, 21-Nov-2019.)
𝐴 ∈ V    &   (π‘₯ = 𝐴 β†’ (πœ“ β†’ πœ‘))    β‡’   (πœ“ β†’ 𝐴 ∈ {π‘₯ ∣ πœ‘})
 
Theoremelabg2 14540* One implication of elabg 2884. (Contributed by BJ, 21-Nov-2019.)
(π‘₯ = 𝐴 β†’ (πœ“ β†’ πœ‘))    β‡’   (𝐴 ∈ 𝑉 β†’ (πœ“ β†’ 𝐴 ∈ {π‘₯ ∣ πœ‘}))
 
Theorembj-rspgt 14541 Restricted specialization, generalized. Weakens a hypothesis of rspccv 2839 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    &   β„²π‘₯πœ“    β‡’   (βˆ€π‘₯(π‘₯ = 𝐴 β†’ (πœ‘ β†’ πœ“)) β†’ (βˆ€π‘₯ ∈ 𝐡 πœ‘ β†’ (𝐴 ∈ 𝐡 β†’ πœ“)))
 
Theorembj-rspg 14542 Restricted specialization, generalized. Weakens a hypothesis of rspccv 2839 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    &   β„²π‘₯πœ“    &   (π‘₯ = 𝐴 β†’ (πœ‘ β†’ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐡 πœ‘ β†’ (𝐴 ∈ 𝐡 β†’ πœ“))
 
Theoremcbvrald 14543* Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    &   (πœ‘ β†’ β„²π‘¦πœ“)    &   (πœ‘ β†’ β„²π‘₯πœ’)    &   (πœ‘ β†’ (π‘₯ = 𝑦 β†’ (πœ“ ↔ πœ’)))    β‡’   (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐴 πœ“ ↔ βˆ€π‘¦ ∈ 𝐴 πœ’))
 
Theorembj-intabssel 14544 Version of intss1 3860 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
β„²π‘₯𝐴    β‡’   (𝐴 ∈ 𝑉 β†’ ([𝐴 / π‘₯]πœ‘ β†’ ∩ {π‘₯ ∣ πœ‘} βŠ† 𝐴))
 
Theorembj-intabssel1 14545 Version of intss1 3860 using a class abstraction and implicit substitution. Closed form of intmin3 3872. (Contributed by BJ, 29-Nov-2019.)
β„²π‘₯𝐴    &   β„²π‘₯πœ“    &   (π‘₯ = 𝐴 β†’ (πœ“ β†’ πœ‘))    β‡’   (𝐴 ∈ 𝑉 β†’ (πœ“ β†’ ∩ {π‘₯ ∣ πœ‘} βŠ† 𝐴))
 
Theorembj-elssuniab 14546 Version of elssuni 3838 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
β„²π‘₯𝐴    β‡’   (𝐴 ∈ 𝑉 β†’ ([𝐴 / π‘₯]πœ‘ β†’ 𝐴 βŠ† βˆͺ {π‘₯ ∣ πœ‘}))
 
Theorembj-sseq 14547 If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.)
(πœ‘ β†’ (πœ“ ↔ 𝐴 βŠ† 𝐡))    &   (πœ‘ β†’ (πœ’ ↔ 𝐡 βŠ† 𝐴))    β‡’   (πœ‘ β†’ ((πœ“ ∧ πœ’) ↔ 𝐴 = 𝐡))
 
12.2.5  Decidability of classes

The question of decidability is essential in intuitionistic logic. In intuitionistic set theories, it is natural to define decidability of a set (or class) as decidability of membership in it. One can parameterize this notion with another set (or class) since it is often important to assess decidability of membership in one class among elements of another class. Namely, one will say that "𝐴 is decidable in 𝐡 " if βˆ€π‘₯ ∈ 𝐡DECID π‘₯ ∈ 𝐴 (see df-dcin 14549).

Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 14596).

 
Syntaxwdcin 14548 Syntax for decidability of a class in another.
wff 𝐴 DECIDin 𝐡
 
Definitiondf-dcin 14549* Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)
(𝐴 DECIDin 𝐡 ↔ βˆ€π‘₯ ∈ 𝐡 DECID π‘₯ ∈ 𝐴)
 
Theoremdecidi 14550 Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
(𝐴 DECIDin 𝐡 β†’ (𝑋 ∈ 𝐡 β†’ (𝑋 ∈ 𝐴 ∨ Β¬ 𝑋 ∈ 𝐴)))
 
Theoremdecidr 14551* Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
(πœ‘ β†’ (π‘₯ ∈ 𝐡 β†’ (π‘₯ ∈ 𝐴 ∨ Β¬ π‘₯ ∈ 𝐴)))    β‡’   (πœ‘ β†’ 𝐴 DECIDin 𝐡)
 
Theoremdecidin 14552 If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.)
(πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   (πœ‘ β†’ 𝐴 DECIDin 𝐡)    &   (πœ‘ β†’ 𝐡 DECIDin 𝐢)    β‡’   (πœ‘ β†’ 𝐴 DECIDin 𝐢)
 
Theoremuzdcinzz 14553 An upperset of integers is decidable in the integers. Reformulation of eluzdc 9610. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)
(𝑀 ∈ β„€ β†’ (β„€β‰₯β€˜π‘€) DECIDin β„€)
 
Theoremsumdc2 14554* Alternate proof of sumdc 11366, without disjoint variable condition on 𝑁, π‘₯ (longer because the statement is taylored to the proof sumdc 11366). (Contributed by BJ, 19-Feb-2022.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ (β„€β‰₯β€˜π‘€)DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ DECID 𝑁 ∈ 𝐴)
 
12.2.6  Disjoint union
 
Theoremdjucllem 14555* Lemma for djulcl 7050 and djurcl 7051. (Contributed by BJ, 4-Jul-2022.)
𝑋 ∈ V    &   πΉ = (π‘₯ ∈ V ↦ βŸ¨π‘‹, π‘₯⟩)    β‡’   (𝐴 ∈ 𝐡 β†’ ((𝐹 β†Ύ 𝐡)β€˜π΄) ∈ ({𝑋} Γ— 𝐡))
 
TheoremdjulclALT 14556 Shortening of djulcl 7050 using djucllem 14555. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐢 ∈ 𝐴 β†’ ((inl β†Ύ 𝐴)β€˜πΆ) ∈ (𝐴 βŠ” 𝐡))
 
TheoremdjurclALT 14557 Shortening of djurcl 7051 using djucllem 14555. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐢 ∈ 𝐡 β†’ ((inr β†Ύ 𝐡)β€˜πΆ) ∈ (𝐴 βŠ” 𝐡))
 
12.2.7  Miscellaneous
 
Theoremfunmptd 14558 The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5254, then prove funmptd 14558 from it, and then prove funmpt 5255 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)

(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡))    β‡’   (πœ‘ β†’ Fun 𝐹)
 
Theoremfnmptd 14559* The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝐹 Fn 𝐴)
 
Theoremif0ab 14560* Expression of a conditional class as a class abstraction when the False alternative is the empty class: in that case, the conditional class is the extension, in the True alternative, of the condition.

Remark: a consequence which could be formalized is the inclusion if(πœ‘, 𝐴, βˆ…) βŠ† 𝐴 and therefore, using elpwg 3584, (𝐴 ∈ 𝑉 β†’ if(πœ‘, 𝐴, βˆ…) ∈ 𝒫 𝐴), from which fmelpw1o 14561 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

if(πœ‘, 𝐴, βˆ…) = {π‘₯ ∈ 𝐴 ∣ πœ‘}
 
Theoremfmelpw1o 14561 With a formula πœ‘ one can associate an element of 𝒫 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than ⊀ and βŠ₯, by nndc 851, which translate to 1o and βˆ… respectively by iftrue 3540 and iffalse 3543, giving pwtrufal 14750).

As proved in if0ab 14560, the associated element of 𝒫 1o is the extension, in 𝒫 1o, of the formula πœ‘. (Contributed by BJ, 15-Aug-2024.)

if(πœ‘, 1o, βˆ…) ∈ 𝒫 1o
 
Theorembj-charfun 14562* Properties of the characteristic function on the class 𝑋 of the class 𝐴. (Contributed by BJ, 15-Aug-2024.)
(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)))    β‡’   (πœ‘ β†’ ((𝐹:π‘‹βŸΆπ’« 1o ∧ (𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))):((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))⟢2o) ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…)))
 
Theorembj-charfundc 14563* Properties of the characteristic function on the class 𝑋 of the class 𝐴, provided membership in 𝐴 is decidable in 𝑋. (Contributed by BJ, 6-Aug-2024.)
(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ 𝐴)    β‡’   (πœ‘ β†’ (𝐹:π‘‹βŸΆ2o ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…)))
 
Theorembj-charfundcALT 14564* Alternate proof of bj-charfundc 14563. It was expected to be much shorter since it uses bj-charfun 14562 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ 𝐴)    β‡’   (πœ‘ β†’ (𝐹:π‘‹βŸΆ2o ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…)))
 
Theorembj-charfunr 14565* If a class 𝐴 has a "weak" characteristic function on a class 𝑋, then negated membership in 𝐴 is decidable (in other words, membership in 𝐴 is testable) in 𝑋.

The hypothesis imposes that 𝑋 be a set. As usual, it could be formulated as (πœ‘ β†’ (𝐹:π‘‹βŸΆΟ‰ ∧ ...)) to deal with general classes, but that extra generality would not make the theorem much more useful.

The theorem would still hold if the codomain of 𝑓 were any class with testable equality to the point where (𝑋 βˆ– 𝐴) is sent. (Contributed by BJ, 6-Aug-2024.)

(πœ‘ β†’ βˆƒπ‘“ ∈ (Ο‰ β†‘π‘š 𝑋)(βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(π‘“β€˜π‘₯) β‰  βˆ… ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(π‘“β€˜π‘₯) = βˆ…))    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 DECID Β¬ π‘₯ ∈ 𝐴)
 
Theorembj-charfunbi 14566* In an ambient set 𝑋, if membership in 𝐴 is stable, then it is decidable if and only if 𝐴 has a characteristic function.

This characterization can be applied to singletons when the set 𝑋 has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.)

(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 STAB π‘₯ ∈ 𝐴)    β‡’   (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ 𝐴 ↔ βˆƒπ‘“ ∈ (2o β†‘π‘š 𝑋)(βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(π‘“β€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(π‘“β€˜π‘₯) = βˆ…)))
 
12.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes

This section develops constructive Zermelo--Fraenkel set theory (CZF) on top of intuitionistic logic. It is a constructive theory in the sense that its logic is intuitionistic and it is predicative. "Predicative" means that new sets can be constructed only from already constructed sets. In particular, the axiom of separation ax-sep 4122 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 14639. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4119 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 14737 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 14696. Similarly, the axiom of powerset ax-pow 4175 is not predicative (checking whether a set is included in another requires to universally quantifier over that "not yet constructed" set) and is replaced in CZF by the axiom of fullness or the axiom of subset collection ax-sscoll 14742.

In an intuitionistic context, the axiom of regularity is stated in IZF as well as in CZF as the axiom of set induction ax-setind 4537. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 14723.

For more details on CZF, a useful set of notes is

Peter Aczel and Michael Rathjen, CST Book draft. (available at http://www1.maths.leeds.ac.uk/~rathjen/book.pdf 14723)

and an interesting article is

Michael Shulman, Comparing material and structural set theories, Annals of Pure and Applied Logic, Volume 170, Issue 4 (Apr. 2019), 465--504. https://doi.org/10.48550/arXiv.1808.05204 14723

I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.

 
12.2.8.1  Bounded formulas

The present definition of bounded formulas emerged from a discussion on GitHub between Jim Kingdon, Mario Carneiro and I, started 23-Sept-2019 (see https://github.com/metamath/set.mm/issues/1173 and links therein).

In order to state certain axiom schemes of Constructive Zermelo–Fraenkel (CZF) set theory, like the axiom scheme of bounded (or restricted, or Ξ”0) separation, it is necessary to distinguish certain formulas, called bounded (or restricted, or Ξ”0) formulas. The necessity of considering bounded formulas also arises in several theories of bounded arithmetic, both classical or intuitionistic, for instance to state the axiom scheme of Ξ”0-induction.

To formalize this in Metamath, there are several choices to make.

A first choice is to either create a new type for bounded formulas, or to create a predicate on formulas that indicates whether they are bounded. In the first case, one creates a new type "wff0" with a new set of metavariables (ph0 ...) and an axiom "$a wff ph0 " ensuring that bounded formulas are formulas, so that one can reuse existing theorems, and then axioms take the form "$a wff0 ( ph0 -> ps0 )", etc. In the second case, one introduces a predicate "BOUNDED " with the intended meaning that "BOUNDED πœ‘ " is a formula meaning that πœ‘ is a bounded formula. We choose the second option, since the first would complicate the grammar, risking to make it ambiguous. (TODO: elaborate.)

A second choice is to view "bounded" either as a syntactic or a semantic property. For instance, βˆ€π‘₯⊀ is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to ⊀ which is bounded. We choose the second option, so that formulas using defined symbols can be proved bounded.

A third choice is in the form of the axioms, either in closed form or in inference form. One cannot state all the axioms in closed form, especially ax-bd0 14568. Indeed, if we posited it in closed form, then we could prove for instance (πœ‘ β†’ BOUNDED πœ‘) and (Β¬ πœ‘ β†’ BOUNDED πœ‘) which is problematic (with the law of excluded middle, this would entail that all formulas are bounded, but even without it, too many formulas could be proved bounded...). (TODO: elaborate.)

Having ax-bd0 14568 in inference form ensures that a formula can be proved bounded only if it is equivalent *for all values of the free variables* to a syntactically bounded one. The other axioms (ax-bdim 14569 through ax-bdsb 14577) can be written either in closed or inference form. The fact that ax-bd0 14568 is an inference is enough to ensure that the closed forms cannot be "exploited" to prove that some unbounded formulas are bounded. (TODO: check.) However, we state all the axioms in inference form to make it clear that we do not exploit any over-permissiveness.

Finally, note that our logic has no terms, only variables. Therefore, we cannot prove for instance that π‘₯ ∈ Ο‰ is a bounded formula. However, since Ο‰ can be defined as "the 𝑦 such that PHI" a proof using the fact that π‘₯ ∈ Ο‰ is bounded can be converted to a proof in iset.mm by replacing Ο‰ with 𝑦 everywhere and prepending the antecedent PHI, since π‘₯ ∈ 𝑦 is bounded by ax-bdel 14576. For a similar method, see bj-omtrans 14711.

Note that one cannot add an axiom BOUNDED π‘₯ ∈ 𝐴 since by bdph 14605 it would imply that every formula is bounded.

 
Syntaxwbd 14567 Syntax for the predicate BOUNDED.
wff BOUNDED πœ‘
 
Axiomax-bd0 14568 If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)
(πœ‘ ↔ πœ“)    β‡’   (BOUNDED πœ‘ β†’ BOUNDED πœ“)
 
Axiomax-bdim 14569 An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED πœ‘    &   BOUNDED πœ“    β‡’   BOUNDED (πœ‘ β†’ πœ“)
 
Axiomax-bdan 14570 The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED πœ‘    &   BOUNDED πœ“    β‡’   BOUNDED (πœ‘ ∧ πœ“)
 
Axiomax-bdor 14571 The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED πœ‘    &   BOUNDED πœ“    β‡’   BOUNDED (πœ‘ ∨ πœ“)
 
Axiomax-bdn 14572 The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED πœ‘    β‡’   BOUNDED Β¬ πœ‘
 
Axiomax-bdal 14573* A bounded universal quantification of a bounded formula is bounded. Note the disjoint variable condition on π‘₯, 𝑦. (Contributed by BJ, 25-Sep-2019.)
BOUNDED πœ‘    β‡’   BOUNDED βˆ€π‘₯ ∈ 𝑦 πœ‘
 
Axiomax-bdex 14574* A bounded existential quantification of a bounded formula is bounded. Note the disjoint variable condition on π‘₯, 𝑦. (Contributed by BJ, 25-Sep-2019.)
BOUNDED πœ‘    β‡’   BOUNDED βˆƒπ‘₯ ∈ 𝑦 πœ‘
 
Axiomax-bdeq 14575 An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
BOUNDED π‘₯ = 𝑦
 
Axiomax-bdel 14576 An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
BOUNDED π‘₯ ∈ 𝑦
 
Axiomax-bdsb 14577 A formula resulting from proper substitution in a bounded formula is bounded. This probably cannot be proved from the other axioms, since neither the definiens in df-sb 1763, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED πœ‘    β‡’   BOUNDED [𝑦 / π‘₯]πœ‘
 
Theorembdeq 14578 Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
(πœ‘ ↔ πœ“)    β‡’   (BOUNDED πœ‘ ↔ BOUNDED πœ“)
 
Theorembd0 14579 A formula equivalent to a bounded one is bounded. See also bd0r 14580. (Contributed by BJ, 3-Oct-2019.)
BOUNDED πœ‘    &   (πœ‘ ↔ πœ“)    β‡’   BOUNDED πœ“
 
Theorembd0r 14580 A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 14579) biconditional in the hypothesis, to work better with definitions (πœ“ is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)
BOUNDED πœ‘    &   (πœ“ ↔ πœ‘)    β‡’   BOUNDED πœ“
 
Theorembdbi 14581 A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED πœ‘    &   BOUNDED πœ“    β‡’   BOUNDED (πœ‘ ↔ πœ“)
 
Theorembdstab 14582 Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED πœ‘    β‡’   BOUNDED STAB πœ‘
 
Theorembddc 14583 Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED πœ‘    β‡’   BOUNDED DECID πœ‘
 
Theorembd3or 14584 A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED πœ‘    &   BOUNDED πœ“    &   BOUNDED πœ’    β‡’   BOUNDED (πœ‘ ∨ πœ“ ∨ πœ’)
 
Theorembd3an 14585 A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED πœ‘    &   BOUNDED πœ“    &   BOUNDED πœ’    β‡’   BOUNDED (πœ‘ ∧ πœ“ ∧ πœ’)
 
Theorembdth 14586 A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
πœ‘    β‡’   BOUNDED πœ‘
 
Theorembdtru 14587 The truth value ⊀ is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED ⊀
 
Theorembdfal 14588 The truth value βŠ₯ is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED βŠ₯
 
Theorembdnth 14589 A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
Β¬ πœ‘    β‡’   BOUNDED πœ‘
 
TheorembdnthALT 14590 Alternate proof of bdnth 14589 not using bdfal 14588. Then, bdfal 14588 can be proved from this theorem, using fal 1360. The total number of proof steps would be 17 (for bdnthALT 14590) + 3 = 20, which is more than 8 (for bdfal 14588) + 9 (for bdnth 14589) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Β¬ πœ‘    β‡’   BOUNDED πœ‘
 
Theorembdxor 14591 The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED πœ‘    &   BOUNDED πœ“    β‡’   BOUNDED (πœ‘ ⊻ πœ“)
 
Theorembj-bdcel 14592* Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
BOUNDED 𝑦 = 𝐴    β‡’   BOUNDED 𝐴 ∈ π‘₯
 
Theorembdab 14593 Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
BOUNDED πœ‘    β‡’   BOUNDED π‘₯ ∈ {𝑦 ∣ πœ‘}
 
Theorembdcdeq 14594 Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED πœ‘    β‡’   BOUNDED CondEq(π‘₯ = 𝑦 β†’ πœ‘)
 
12.2.8.2  Bounded classes

In line with our definitions of classes as extensions of predicates, it is useful to define a predicate for bounded classes, which is done in df-bdc 14596. Note that this notion is only a technical device which can be used to shorten proofs of (semantic) boundedness of formulas.

As will be clear by the end of this subsection (see for instance bdop 14630), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, BOUNDED πœ‘ β‡’ BOUNDED ⟨{π‘₯ ∣ πœ‘}, ({𝑦, suc 𝑧} Γ— βŸ¨π‘‘, βˆ…βŸ©)⟩. The proofs are long since one has to prove boundedness at each step of the construction, without being able to prove general theorems like BOUNDED 𝐴 β‡’ BOUNDED {𝐴}.

 
Syntaxwbdc 14595 Syntax for the predicate BOUNDED.
wff BOUNDED 𝐴
 
Definitiondf-bdc 14596* Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
(BOUNDED 𝐴 ↔ βˆ€π‘₯BOUNDED π‘₯ ∈ 𝐴)
 
Theorembdceq 14597 Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
𝐴 = 𝐡    β‡’   (BOUNDED 𝐴 ↔ BOUNDED 𝐡)
 
Theorembdceqi 14598 A class equal to a bounded one is bounded. Note the use of ax-ext 2159. See also bdceqir 14599. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   π΄ = 𝐡    β‡’   BOUNDED 𝐡
 
Theorembdceqir 14599 A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 14598) equality in the hypothesis, to work better with definitions (𝐡 is the definiendum that one wants to prove bounded; see comment of bd0r 14580). (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   π΅ = 𝐴    β‡’   BOUNDED 𝐡
 
Theorembdel 14600* The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
(BOUNDED 𝐴 β†’ BOUNDED π‘₯ ∈ 𝐴)
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