P. 165 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16401-16500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-fadc 16401	A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
⊢ (¬ 𝜑 → DECID 𝜑)
Theorem	bj-dcfal 16402	The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
⊢ DECID ⊥
Theorem	bj-dcstab 16403	A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
⊢ (DECID 𝜑 → STAB 𝜑)
Theorem	bj-nnbidc 16404	If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 16391. (Contributed by BJ, 24-Nov-2023.)
⊢ (¬ ¬ 𝜑 → (DECID 𝜑 ↔ 𝜑))
Theorem	bj-nndcALT 16405	Alternate proof of nndc 858. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)
⊢ ¬ ¬ DECID 𝜑
Theorem	bj-dcdc 16406	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 𝜑 ↔ DECID 𝜑)
Theorem	bj-stdc 16407	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 𝜑 ↔ DECID 𝜑)
Theorem	bj-dcst 16408	Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)
⊢ (DECID STAB 𝜑 ↔ STAB 𝜑)
14.3.2  Predicate calculus
Theorem	bj-ex 16409*	Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1646 and 19.9ht 1689 or 19.23ht 1545). (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → 𝜑)
Theorem	bj-hbalt 16410	Closed form of hbal 1525 (copied from set.mm). (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
Theorem	bj-nfalt 16411	Closed form of nfal 1624 (copied from set.mm). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)
Theorem	spimd 16412	Deduction form of spim 1786. (Contributed by BJ, 17-Oct-2019.)
⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	2spim 16413*	Double substitution, as in spim 1786. (Contributed by BJ, 17-Oct-2019.)
⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑧𝜒    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑧∀𝑥𝜓 → 𝜒)
Theorem	ch2var 16414*	Implicit substitution of 𝑦 for 𝑥 and 𝑡 for 𝑧 into a theorem. (Contributed by BJ, 17-Oct-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	ch2varv 16415*	Version of ch2var 16414 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	bj-exlimmp 16416	Lemma for bj-vtoclgf 16423. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓))
Theorem	bj-exlimmpi 16417	Lemma for bj-vtoclgf 16423. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    &   ⊢ (𝜒 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-sbimedh 16418	A strengthening of sbiedh 1835 (same proof). (Contributed by BJ, 16-Dec-2019.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒))
Theorem	bj-sbimeh 16419	A strengthening of sbieh 1838 (same proof). (Contributed by BJ, 16-Dec-2019.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)
Theorem	bj-sbime 16420	A strengthening of sbie 1839 (same proof). (Contributed by BJ, 16-Dec-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)
14.3.3  Set theorey miscellaneous
Theorem	bj-el2oss1o 16421	Shorter proof of el2oss1o 6611 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o)
14.3.4  Extensionality Various utility theorems using FOL and extensionality.
Theorem	bj-vtoclgft 16422	Weakening two hypotheses of vtoclgf 2862. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓))
Theorem	bj-vtoclgf 16423	Weakening two hypotheses of vtoclgf 2862. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)
Theorem	elabgf0 16424	Lemma for elabgf 2948. (Contributed by BJ, 21-Nov-2019.)
⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
Theorem	elabgft1 16425	One implication of elabgf 2948, in closed form. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))
Theorem	elabgf1 16426	One implication of elabgf 2948. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
Theorem	elabgf2 16427	One implication of elabgf 2948. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
Theorem	elabf1 16428*	One implication of elabf 2949. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
Theorem	elabf2 16429*	One implication of elabf 2949. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})
Theorem	elab1 16430*	One implication of elab 2950. (Contributed by BJ, 21-Nov-2019.)
⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
Theorem	elab2a 16431*	One implication of elab 2950. (Contributed by BJ, 21-Nov-2019.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})
Theorem	elabg2 16432*	One implication of elabg 2952. (Contributed by BJ, 21-Nov-2019.)
⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
Theorem	bj-rspgt 16433	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2907 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)))
Theorem	bj-rspg 16434	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2907 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))
Theorem	cbvrald 16435*	Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))
Theorem	bj-intabssel 16436	Version of intss1 3943 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))
Theorem	bj-intabssel1 16437	Version of intss1 3943 using a class abstraction and implicit substitution. Closed form of intmin3 3955. (Contributed by BJ, 29-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))
Theorem	bj-elssuniab 16438	Version of elssuni 3921 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → 𝐴 ⊆ ∪ {𝑥 ∣ 𝜑}))
Theorem	bj-sseq 16439	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.)
⊢ (𝜑 → (𝜓 ↔ 𝐴 ⊆ 𝐵))    &   ⊢ (𝜑 → (𝜒 ↔ 𝐵 ⊆ 𝐴))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝐴 = 𝐵))
14.3.5  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 𝐵 " if ∀𝑥 ∈ 𝐵DECID 𝑥 ∈ 𝐴 (see df-dcin 16441). Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 16487).
Syntax	wdcin 16440	Syntax for decidability of a class in another.
wff 𝐴 DECIDin 𝐵
Definition	df-dcin 16441*	Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)
⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)
Theorem	decidi 16442	Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
⊢ (𝐴 DECIDin 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))
Theorem	decidr 16443*	Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))    ⇒   ⊢ (𝜑 → 𝐴 DECIDin 𝐵)
Theorem	decidin 16444	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.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 DECIDin 𝐵)    &   ⊢ (𝜑 → 𝐵 DECIDin 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 DECIDin 𝐶)
Theorem	uzdcinzz 16445	An upperset of integers is decidable in the integers. Reformulation of eluzdc 9844. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)
⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) DECIDin ℤ)
Theorem	sumdc2 16446*	Alternate proof of sumdc 11936, without disjoint variable condition on 𝑁, 𝑥 (longer because the statement is taylored to the proof sumdc 11936). (Contributed by BJ, 19-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴)
14.3.6  Disjoint union
Theorem	djucllem 16447*	Lemma for djulcl 7250 and djurcl 7251. (Contributed by BJ, 4-Jul-2022.)
⊢ 𝑋 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)    ⇒   ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))
Theorem	djulclALT 16448	Shortening of djulcl 7250 using djucllem 16447. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djurclALT 16449	Shortening of djurcl 7251 using djucllem 16447. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵))
14.3.7  Miscellaneous
Theorem	funmptd 16450	The maps-to notation defines a function (deduction form). Note: one should similarly prove a deduction form of funopab4 5363, then prove funmptd 16450 from it, and then prove funmpt 5364 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	fnmptd 16451*	The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 Fn 𝐴)
Theorem	if0ab 16452*	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(𝜑, 𝐴, ∅) ⊆ 𝐴 and therefore, using elpwg 3660, ⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 7465 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)
⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	bj-charfun 16453*	Properties of the characteristic function on the class 𝑋 of the class 𝐴. (Contributed by BJ, 15-Aug-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    ⇒   ⊢ (𝜑 → ((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2o) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
Theorem	bj-charfundc 16454*	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.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
Theorem	bj-charfundcALT 16455*	Alternate proof of bj-charfundc 16454. It was expected to be much shorter since it uses bj-charfun 16453 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.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
Theorem	bj-charfunr 16456*	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 ⊢ (𝜑 → (𝐹:𝑋⟶ω ∧ ...)) 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 ∖ 𝐴) is sent. (Contributed by BJ, 6-Aug-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (ω ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴)
Theorem	bj-charfunbi 16457*	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 STAB 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)))
14.3.8  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 4207 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 16530. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4204 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 16628 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 16587. Similarly, the axiom of powerset ax-pow 4264 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 16633. 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 4635. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 16614. 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 16614) 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 16614 I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.
14.3.8.1  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 Δ0) separation, it is necessary to distinguish certain formulas, called bounded (or restricted, or Δ0) 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 Δ0-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 𝜑 " is a formula meaning that 𝜑 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, ∀𝑥⊤ 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 16459. Indeed, if we posited it in closed form, then we could prove for instance ⊢ (𝜑 → BOUNDED 𝜑) and ⊢ (¬ 𝜑 → BOUNDED 𝜑) 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 16459 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 16460 through ax-bdsb 16468) can be written either in closed or inference form. The fact that ax-bd0 16459 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 𝑥 ∈ ω is a bounded formula. However, since ω can be defined as "the 𝑦 such that PHI" a proof using the fact that 𝑥 ∈ ω is bounded can be converted to a proof in iset.mm by replacing ω with 𝑦 everywhere and prepending the antecedent PHI, since 𝑥 ∈ 𝑦 is bounded by ax-bdel 16467. For a similar method, see bj-omtrans 16602. Note that one cannot add an axiom ⊢ BOUNDED 𝑥 ∈ 𝐴 since by bdph 16496 it would imply that every formula is bounded.
Syntax	wbd 16458	Syntax for the predicate BOUNDED.
wff BOUNDED 𝜑
Axiom	ax-bd0 16459	If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
Axiom	ax-bdim 16460	An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 → 𝜓)
Axiom	ax-bdan 16461	The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ∧ 𝜓)
Axiom	ax-bdor 16462	The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ∨ 𝜓)
Axiom	ax-bdn 16463	The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ¬ 𝜑
Axiom	ax-bdal 16464*	A bounded universal quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝜑
Axiom	ax-bdex 16465*	A bounded existential quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
Axiom	ax-bdeq 16466	An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 = 𝑦
Axiom	ax-bdel 16467	An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 ∈ 𝑦
Axiom	ax-bdsb 16468	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 1811, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdeq 16469	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓)
Theorem	bd0 16470	A formula equivalent to a bounded one is bounded. See also bd0r 16471. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ BOUNDED 𝜓
Theorem	bd0r 16471	A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16470) biconditional in the hypothesis, to work better with definitions (𝜓 is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝜓 ↔ 𝜑)    ⇒   ⊢ BOUNDED 𝜓
Theorem	bdbi 16472	A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ↔ 𝜓)
Theorem	bdstab 16473	Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED STAB 𝜑
Theorem	bddc 16474	Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED DECID 𝜑
Theorem	bd3or 16475	A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    &   ⊢ BOUNDED 𝜒    ⇒   ⊢ BOUNDED (𝜑 ∨ 𝜓 ∨ 𝜒)
Theorem	bd3an 16476	A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    &   ⊢ BOUNDED 𝜒    ⇒   ⊢ BOUNDED (𝜑 ∧ 𝜓 ∧ 𝜒)
Theorem	bdth 16477	A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
⊢ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdtru 16478	The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ⊤
Theorem	bdfal 16479	The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ⊥
Theorem	bdnth 16480	A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
⊢ ¬ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdnthALT 16481	Alternate proof of bdnth 16480 not using bdfal 16479. Then, bdfal 16479 can be proved from this theorem, using fal 1404. The total number of proof steps would be 17 (for bdnthALT 16481) + 3 = 20, which is more than 8 (for bdfal 16479) + 9 (for bdnth 16480) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdxor 16482	The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ⊻ 𝜓)
Theorem	bj-bdcel 16483*	Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
⊢ BOUNDED 𝑦 = 𝐴    ⇒   ⊢ BOUNDED 𝐴 ∈ 𝑥
Theorem	bdab 16484	Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑}
Theorem	bdcdeq 16485	Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED CondEq(𝑥 = 𝑦 → 𝜑)
14.3.8.2  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 16487. 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 16521), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, ⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ⟨{𝑥 ∣ 𝜑}, ({𝑦, suc 𝑧} × ⟨𝑡, ∅⟩)⟩. 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 ⇒ ⊢ BOUNDED {𝐴}.
Syntax	wbdc 16486	Syntax for the predicate BOUNDED.
wff BOUNDED 𝐴
Definition	df-bdc 16487*	Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
Theorem	bdceq 16488	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)
Theorem	bdceqi 16489	A class equal to a bounded one is bounded. Note the use of ax-ext 2213. See also bdceqir 16490. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ BOUNDED 𝐵
Theorem	bdceqir 16490	A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16489) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 16471). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 = 𝐴    ⇒   ⊢ BOUNDED 𝐵
Theorem	bdel 16491*	The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ (BOUNDED 𝐴 → BOUNDED 𝑥 ∈ 𝐴)
Theorem	bdeli 16492*	Inference associated with bdel 16491. Its converse is bdelir 16493. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ∈ 𝐴
Theorem	bdelir 16493*	Inference associated with df-bdc 16487. Its converse is bdeli 16492. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 ∈ 𝐴    ⇒   ⊢ BOUNDED 𝐴
Theorem	bdcv 16494	A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥
Theorem	bdcab 16495	A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED {𝑥 ∣ 𝜑}
Theorem	bdph 16496	A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
⊢ BOUNDED {𝑥 ∣ 𝜑}    ⇒   ⊢ BOUNDED 𝜑
Theorem	bds 16497*	Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 16468; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 16468. (Contributed by BJ, 19-Nov-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ BOUNDED 𝜓
Theorem	bdcrab 16498*	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 {𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	bdne 16499	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝑥 ≠ 𝑦
Theorem	bdnel 16500*	Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ∉ 𝐴