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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 2spim 13801* | Double substitution, as in spim 1731. (Contributed by BJ, 17-Oct-2019.) |
⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑧𝜒 & ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒)) ⇒ ⊢ (∀𝑧∀𝑥𝜓 → 𝜒) | ||
Theorem | ch2var 13802* | 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 13803* | Version of ch2var 13802 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 13804 | Lemma for bj-vtoclgf 13811. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝜒 → 𝜑) ⇒ ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓)) | ||
Theorem | bj-exlimmpi 13805 | Lemma for bj-vtoclgf 13811. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝜒 → 𝜑) & ⊢ (𝜒 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥𝜒 → 𝜓) | ||
Theorem | bj-sbimedh 13806 | A strengthening of sbiedh 1780 (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 13807 | A strengthening of sbieh 1783 (same proof). (Contributed by BJ, 16-Dec-2019.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓) | ||
Theorem | bj-sbime 13808 | A strengthening of sbie 1784 (same proof). (Contributed by BJ, 16-Dec-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓) | ||
Theorem | bj-el2oss1o 13809 | Shorter proof of el2oss1o 6422 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o) | ||
Various utility theorems using FOL and extensionality. | ||
Theorem | bj-vtoclgft 13810 | Weakening two hypotheses of vtoclgf 2788. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓)) | ||
Theorem | bj-vtoclgf 13811 | Weakening two hypotheses of vtoclgf 2788. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → 𝜑) & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓) | ||
Theorem | elabgf0 13812 | Lemma for elabgf 2872. (Contributed by BJ, 21-Nov-2019.) |
⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) | ||
Theorem | elabgft1 13813 | One implication of elabgf 2872, in closed form. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)) | ||
Theorem | elabgf1 13814 | One implication of elabgf 2872. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) | ||
Theorem | elabgf2 13815 | One implication of elabgf 2872. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | ||
Theorem | elabf1 13816* | One implication of elabf 2873. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) | ||
Theorem | elabf2 13817* | One implication of elabf 2873. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | elab1 13818* | One implication of elab 2874. (Contributed by BJ, 21-Nov-2019.) |
⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) | ||
Theorem | elab2a 13819* | One implication of elab 2874. (Contributed by BJ, 21-Nov-2019.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | elabg2 13820* | One implication of elabg 2876. (Contributed by BJ, 21-Nov-2019.) |
⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | ||
Theorem | bj-rspgt 13821 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2831 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 13822 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2831 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 13823* | 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 13824 | Version of intss1 3846 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 13825 | Version of intss1 3846 using a class abstraction and implicit substitution. Closed form of intmin3 3858. (Contributed by BJ, 29-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)) | ||
Theorem | bj-elssuniab 13826 | Version of elssuni 3824 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 13827 | 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)) | ||
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 13829). Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 13876). | ||
Syntax | wdcin 13828 | Syntax for decidability of a class in another. |
wff 𝐴 DECIDin 𝐵 | ||
Definition | df-dcin 13829* | Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.) |
⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) | ||
Theorem | decidi 13830 | Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.) |
⊢ (𝐴 DECIDin 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴))) | ||
Theorem | decidr 13831* | 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 13832 | 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 13833 | An upperset of integers is decidable in the integers. Reformulation of eluzdc 9569. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.) |
⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) DECIDin ℤ) | ||
Theorem | sumdc2 13834* | Alternate proof of sumdc 11321, without disjoint variable condition on 𝑁, 𝑥 (longer because the statement is taylored to the proof sumdc 11321). (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) | ||
Theorem | djucllem 13835* | Lemma for djulcl 7028 and djurcl 7029. (Contributed by BJ, 4-Jul-2022.) |
⊢ 𝑋 ∈ V & ⊢ 𝐹 = (𝑥 ∈ V ↦ 〈𝑋, 𝑥〉) ⇒ ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵)) | ||
Theorem | djulclALT 13836 | Shortening of djulcl 7028 using djucllem 13835. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) | ||
Theorem | djurclALT 13837 | Shortening of djurcl 7029 using djucllem 13835. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) | ||
Theorem | funmptd 13838 |
The maps-to notation defines a function (deduction form).
Note: one should similarly prove a deduction form of funopab4 5235, then prove funmptd 13838 from it, and then prove funmpt 5236 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | fnmptd 13839* | 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 13840* |
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 3574, ⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 13841 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.) |
⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑} | ||
Theorem | fmelpw1o 13841 |
With a formula 𝜑 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 ⊥, by
nndc 846, which translate to 1o and ∅
respectively by iftrue 3531
and iffalse 3534, giving pwtrufal 14030).
As proved in if0ab 13840, the associated element of 𝒫 1o is the extension, in 𝒫 1o, of the formula 𝜑. (Contributed by BJ, 15-Aug-2024.) |
⊢ if(𝜑, 1o, ∅) ∈ 𝒫 1o | ||
Theorem | bj-charfun 13842* | 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 13843* | 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 13844* | Alternate proof of bj-charfundc 13843. It was expected to be much shorter since it uses bj-charfun 13842 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 13845* |
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 13846* |
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) = ∅))) | ||
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 4107 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 13919. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4104 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 14017 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 13976. Similarly, the axiom of powerset ax-pow 4160 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 14022. 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 4521. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 14003. 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 14003) 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 14003 I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results. | ||
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 13848. 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 13848 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 13849 through ax-bdsb 13857) can be written either in closed or inference form. The fact that ax-bd0 13848 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 13856. For a similar method, see bj-omtrans 13991. Note that one cannot add an axiom ⊢ BOUNDED 𝑥 ∈ 𝐴 since by bdph 13885 it would imply that every formula is bounded. | ||
Syntax | wbd 13847 | Syntax for the predicate BOUNDED. |
wff BOUNDED 𝜑 | ||
Axiom | ax-bd0 13848 | 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 13849 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 → 𝜓) | ||
Axiom | ax-bdan 13850 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ∧ 𝜓) | ||
Axiom | ax-bdor 13851 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ∨ 𝜓) | ||
Axiom | ax-bdn 13852 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ¬ 𝜑 | ||
Axiom | ax-bdal 13853* | 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 13854* | 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 13855 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 = 𝑦 | ||
Axiom | ax-bdel 13856 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 ∈ 𝑦 | ||
Axiom | ax-bdsb 13857 | 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 1756, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdeq 13858 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓) | ||
Theorem | bd0 13859 | A formula equivalent to a bounded one is bounded. See also bd0r 13860. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ BOUNDED 𝜓 | ||
Theorem | bd0r 13860 | A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 13859) 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 13861 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ↔ 𝜓) | ||
Theorem | bdstab 13862 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED STAB 𝜑 | ||
Theorem | bddc 13863 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED DECID 𝜑 | ||
Theorem | bd3or 13864 | 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 13865 | 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 13866 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
⊢ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdtru 13867 | The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ⊤ | ||
Theorem | bdfal 13868 | The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ⊥ | ||
Theorem | bdnth 13869 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
⊢ ¬ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdnthALT 13870 | Alternate proof of bdnth 13869 not using bdfal 13868. Then, bdfal 13868 can be proved from this theorem, using fal 1355. The total number of proof steps would be 17 (for bdnthALT 13870) + 3 = 20, which is more than 8 (for bdfal 13868) + 9 (for bdnth 13869) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdxor 13871 | 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 13872* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) |
⊢ BOUNDED 𝑦 = 𝐴 ⇒ ⊢ BOUNDED 𝐴 ∈ 𝑥 | ||
Theorem | bdab 13873 | 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 13874 | Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED CondEq(𝑥 = 𝑦 → 𝜑) | ||
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 13876. 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 13910), 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 13875 | Syntax for the predicate BOUNDED. |
wff BOUNDED 𝐴 | ||
Definition | df-bdc 13876* | 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 13877 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵) | ||
Theorem | bdceqi 13878 | A class equal to a bounded one is bounded. Note the use of ax-ext 2152. See also bdceqir 13879. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐴 = 𝐵 ⇒ ⊢ BOUNDED 𝐵 | ||
Theorem | bdceqir 13879 | A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 13878) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 13860). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 = 𝐴 ⇒ ⊢ BOUNDED 𝐵 | ||
Theorem | bdel 13880* | 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 13881* | Inference associated with bdel 13880. Its converse is bdelir 13882. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED 𝑥 ∈ 𝐴 | ||
Theorem | bdelir 13882* | Inference associated with df-bdc 13876. Its converse is bdeli 13881. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 ∈ 𝐴 ⇒ ⊢ BOUNDED 𝐴 | ||
Theorem | bdcv 13883 | A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 | ||
Theorem | bdcab 13884 | 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 13885 | 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 13886* | Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 13857; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 13857. (Contributed by BJ, 19-Nov-2019.) |
⊢ BOUNDED 𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ BOUNDED 𝜓 | ||
Theorem | bdcrab 13887* | 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 13888 | Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝑥 ≠ 𝑦 | ||
Theorem | bdnel 13889* | Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED 𝑥 ∉ 𝐴 | ||
Theorem | bdreu 13890* |
Boundedness of existential uniqueness.
Remark regarding restricted quantifiers: the formula ∀𝑥 ∈ 𝐴𝜑 need not be bounded even if 𝐴 and 𝜑 are. Indeed, V is bounded by bdcvv 13892, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 13859, 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 ∃!𝑥 ∈ 𝑦 𝜑 | ||
Theorem | bdrmo 13891* | Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑 | ||
Theorem | bdcvv 13892 | 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 | ||
Theorem | bdsbc 13893 | A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13894. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdsbcALT 13894 | Alternate proof of bdsbc 13893. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdccsb 13895 | 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 / 𝑥⦌𝐴 | ||
Theorem | bdcdif 13896 | The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∖ 𝐵) | ||
Theorem | bdcun 13897 | The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∪ 𝐵) | ||
Theorem | bdcin 13898 | The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∩ 𝐵) | ||
Theorem | bdss 13899 | 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 𝑥 ⊆ 𝐴 | ||
Theorem | bdcnul 13900 | The empty class is bounded. See also bdcnulALT 13901. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ∅ |
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