P. 357 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35601-35700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	satfbrsuc 35601*	The binary relation of a satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 13-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    &   ⊢ 𝑃 = (𝑆‘𝑁)    ⇒   ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})))))
Theorem	satfrel 35602	The value of the satisfaction predicate as function over wff codes at a natural number is a relation. (Contributed by AV, 12-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))
Theorem	satfdmlem 35603*	Lemma for satfdm 35604. (Contributed by AV, 12-Oct-2023.)
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎))))
Theorem	satfdm 35604*	The domain of the satisfaction predicate as function over wff codes does not depend on the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 13-Oct-2023.)
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))
Theorem	satfrnmapom 35605	The range of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is a subset of the power set of all mappings from the natural numbers into the model 𝑀. (Contributed by AV, 13-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀 ↑m ω))
Theorem	satfv0fun 35606	The value of the satisfaction predicate as function over wff codes at ∅ is a function. (Contributed by AV, 15-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
Theorem	satf0 35607*	The satisfaction predicate as function over wff codes in the empty model with an empty binary relation. (Contributed by AV, 14-Sep-2023.)
⊢ (∅ Sat ∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}) ↾ suc ω)
Theorem	satf0sucom 35608*	The satisfaction predicate as function over wff codes in the empty model with an empty binary relation at a successor of ω. (Contributed by AV, 14-Sep-2023.)
⊢ (𝑁 ∈ suc ω → ((∅ Sat ∅)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑁))
Theorem	satf00 35609*	The value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at ∅. (Contributed by AV, 14-Sep-2023.)
⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
Theorem	satf0suclem 35610*	Lemma for satf0suc 35611, sat1el2xp 35614 and fmlasuc0 35619. (Contributed by AV, 19-Sep-2023.)
⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))} ∈ V)
Theorem	satf0suc 35611*	The value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation at a successor. (Contributed by AV, 19-Sep-2023.)
⊢ 𝑆 = (∅ Sat ∅)    ⇒   ⊢ (𝑁 ∈ ω → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}))
Theorem	satf0op 35612*	An element of a value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation expressed as ordered pair. (Contributed by AV, 19-Sep-2023.)
⊢ 𝑆 = (∅ Sat ∅)    ⇒   ⊢ (𝑁 ∈ ω → (𝑋 ∈ (𝑆‘𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘𝑁))))
Theorem	satf0n0 35613	The value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation does not contain the empty set. (Contributed by AV, 19-Sep-2023.)
⊢ (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))
Theorem	sat1el2xp 35614*	The first component of an element of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation is a member of a doubled Cartesian product. (Contributed by AV, 17-Sep-2023.)
⊢ (𝑁 ∈ ω → ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎∃𝑏(1st ‘𝑤) ∈ (ω × (𝑎 × 𝑏)))
Theorem	fmlafv 35615	The valid Godel formulas of height 𝑁 is the domain of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 15-Sep-2023.)
⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
Theorem	fmla 35616	The set of all valid Godel formulas. (Contributed by AV, 20-Sep-2023.)
⊢ (Fmla‘ω) = ∪ 𝑛 ∈ ω (Fmla‘𝑛)
Theorem	fmla0 35617*	The valid Godel formulas of height 0 is the set of all formulas of the form vi ∈ vj ("Godel-set of membership") coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 14-Sep-2023.)
⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
Theorem	fmla0xp 35618	The valid Godel formulas of height 0 is the set of all formulas of the form vi ∈ vj ("Godel-set of membership") coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 15-Sep-2023.)
⊢ (Fmla‘∅) = ({∅} × (ω × ω))
Theorem	fmlasuc0 35619*	The valid Godel formulas of height (𝑁 + 1). (Contributed by AV, 18-Sep-2023.)
⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))}))
Theorem	fmlafvel 35620	A class is a valid Godel formula of height 𝑁 iff it is the first component of a member of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 19-Sep-2023.)
⊢ (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
Theorem	fmlasuc 35621*	The valid Godel formulas of height (𝑁 + 1), expressed by the valid Godel formulas of height 𝑁. (Contributed by AV, 20-Sep-2023.)
⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
Theorem	fmla1 35622*	The valid Godel formulas of height 1 is the set of all formulas of the form (𝑎⊼𝑔𝑏) and ∀𝑔𝑘𝑎 with atoms 𝑎, 𝑏 of the form 𝑥 ∈ 𝑦. (Contributed by AV, 20-Sep-2023.)
⊢ (Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))})
Theorem	isfmlasuc 35623*	The characterization of a Godel formula of height at least 1. (Contributed by AV, 14-Oct-2023.)
⊢ ((𝑁 ∈ ω ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ (Fmla‘suc 𝑁) ↔ (𝐹 ∈ (Fmla‘𝑁) ∨ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))))
Theorem	fmlasssuc 35624	The Godel formulas of height 𝑁 are a subset of the Godel formulas of height 𝑁 + 1. (Contributed by AV, 20-Oct-2023.)
⊢ (𝑁 ∈ ω → (Fmla‘𝑁) ⊆ (Fmla‘suc 𝑁))
Theorem	fmlaomn0 35625	The empty set is not a Godel formula of any height. (Contributed by AV, 21-Oct-2023.)
⊢ (𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))
Theorem	fmlan0 35626	The empty set is not a Godel formula. (Contributed by AV, 19-Nov-2023.)
⊢ ∅ ∉ (Fmla‘ω)
Theorem	gonan0 35627	The "Godel-set of NAND" is a Godel formula of at least height 1. (Contributed by AV, 21-Oct-2023.)
⊢ ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
Theorem	goaln0 35628*	The "Godel-set of universal quantification" is a Godel formula of at least height 1. (Contributed by AV, 22-Oct-2023.)
⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
Theorem	gonarlem 35629*	Lemma for gonar 35630 (induction step). (Contributed by AV, 21-Oct-2023.)
⊢ (𝑁 ∈ ω → (((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁))) → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
Theorem	gonar 35630*	If the "Godel-set of NAND" applied to classes is a Godel formula, the classes are also Godel formulas. Remark: The reverse is not valid for 𝐴 or 𝐵 being of the same height as the "Godel-set of NAND". (Contributed by AV, 21-Oct-2023.)
⊢ ((𝑁 ∈ ω ∧ (𝑎⊼𝑔𝑏) ∈ (Fmla‘𝑁)) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁)))
Theorem	goalrlem 35631*	Lemma for goalr 35632 (induction step). (Contributed by AV, 22-Oct-2023.)
⊢ (𝑁 ∈ ω → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁)) → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁))))
Theorem	goalr 35632*	If the "Godel-set of universal quantification" applied to a class is a Godel formula, the class is also a Godel formula. Remark: The reverse is not valid for 𝐴 being of the same height as the "Godel-set of universal quantification". (Contributed by AV, 22-Oct-2023.)
⊢ ((𝑁 ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑎 ∈ (Fmla‘𝑁))
Theorem	fmla0disjsuc 35633*	The set of valid Godel formulas of height 0 is disjoint with the formulas constructed from Godel-sets for the Sheffer stroke NAND and Godel-set of universal quantification. (Contributed by AV, 20-Oct-2023.)
⊢ ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅
Theorem	fmlasucdisj 35634*	The valid Godel formulas of height (𝑁 + 1) is disjoint with the difference ((Fmla‘suc suc 𝑁) ∖ (Fmla‘suc 𝑁)), expressed by formulas constructed from Godel-sets for the Sheffer stroke NAND and Godel-set of universal quantification based on the valid Godel formulas of height (𝑁 + 1). (Contributed by AV, 20-Oct-2023.)
⊢ (𝑁 ∈ ω → ((Fmla‘suc 𝑁) ∩ {𝑥 ∣ (∃𝑢 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑣 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ∨ ∃𝑢 ∈ (Fmla‘𝑁)∃𝑣 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑢⊼𝑔𝑣))}) = ∅)
Theorem	satfdmfmla 35635	The domain of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is the set of valid Godel formulas of height 𝑁. (Contributed by AV, 13-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))
Theorem	satffunlem 35636	Lemma for satffunlem1lem1 35637 and satffunlem2lem1 35639. (Contributed by AV, 27-Oct-2023.)
⊢ (((Fun 𝑍 ∧ (𝑠 ∈ 𝑍 ∧ 𝑟 ∈ 𝑍) ∧ (𝑢 ∈ 𝑍 ∧ 𝑣 ∈ 𝑍)) ∧ (𝑥 = ((1st ‘𝑠)⊼𝑔(1st ‘𝑟)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑠) ∩ (2nd ‘𝑟)))) ∧ (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣))))) → 𝑦 = 𝑤)
Theorem	satffunlem1lem1 35637*	Lemma for satffunlem1 35642. (Contributed by AV, 17-Oct-2023.)
⊢ (Fun ((𝑀 Sat 𝐸)‘𝑁) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑘 ∈ 𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})
Theorem	satffunlem1lem2 35638*	Lemma 2 for satffunlem1 35642. (Contributed by AV, 23-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}) = ∅)
Theorem	satffunlem2lem1 35639*	Lemma 1 for satffunlem2 35643. (Contributed by AV, 28-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    &   ⊢ 𝐴 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))    &   ⊢ 𝐵 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}    ⇒   ⊢ ((Fun (𝑆‘suc 𝑁) ∧ (𝑆‘𝑁) ⊆ (𝑆‘suc 𝑁)) → Fun {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴))})
Theorem	dmopab3rexdif 35640*	The domain of an ordered pair class abstraction with three nested restricted existential quantifiers with differences. (Contributed by AV, 25-Oct-2023.)
⊢ ((∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) ∧ 𝑆 ⊆ 𝑈) → dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑥 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑥 = 𝐶) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑥 = 𝐴)})
Theorem	satffunlem2lem2 35641*	Lemma 2 for satffunlem2 35643. (Contributed by AV, 27-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    &   ⊢ 𝐴 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))    &   ⊢ 𝐵 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}    ⇒   ⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (dom (𝑆‘suc 𝑁) ∩ dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴))}) = ∅)
Theorem	satffunlem1 35642	Lemma 1 for satffun 35644: induction basis. (Contributed by AV, 28-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))
Theorem	satffunlem2 35643	Lemma 2 for satffun 35644: induction step. (Contributed by AV, 28-Oct-2023.)
⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁)))
Theorem	satffun 35644	The value of the satisfaction predicate as function over wff codes at a natural number is a function. (Contributed by AV, 28-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁))
Theorem	satff 35645	The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 28-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑁):(Fmla‘𝑁)⟶𝒫 (𝑀 ↑m ω))
Theorem	satfun 35646	The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 29-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω))
Theorem	satfvel 35647	An element of the value of the satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at the code 𝑈 for a wff using ∈ , ⊼ , ∀ is a valuation 𝑆:ω⟶𝑀 of the variables (v0 = (𝑆‘∅), v1 = (𝑆‘1o), etc.) so that 𝑈 is true under the assignment 𝑆. (Contributed by AV, 29-Oct-2023.)
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈)) → 𝑆:ω⟶𝑀)
Theorem	satfv0fvfmla0 35648*	The value of the satisfaction predicate as function over a wff code at ∅. (Contributed by AV, 2-Nov-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))})
Theorem	satefv 35649	The simplified satisfaction predicate as function over wff codes in the model 𝑀 at the code 𝑈. (Contributed by AV, 30-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀 Sat∈ 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
Theorem	sate0 35650	The simplified satisfaction predicate for any wff code over an empty model. (Contributed by AV, 6-Oct-2023.) (Revised by AV, 5-Nov-2023.)
⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
Theorem	satef 35651	The simplified satisfaction predicate as function over wff codes over an empty model. (Contributed by AV, 30-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (𝑀 Sat∈ 𝑈)) → 𝑆:ω⟶𝑀)
Theorem	sate0fv0 35652	A simplified satisfaction predicate as function over wff codes over an empty model is an empty set. (Contributed by AV, 31-Oct-2023.)
⊢ (𝑈 ∈ (Fmla‘ω) → (𝑆 ∈ (∅ Sat∈ 𝑈) → 𝑆 = ∅))
Theorem	satefvfmla0 35653*	The simplified satisfaction predicate for wff codes of height 0. (Contributed by AV, 4-Nov-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat∈ 𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋))) ∈ (𝑎‘(2nd ‘(2nd ‘𝑋)))})
Theorem	sategoelfvb 35654	Characterization of a valuation 𝑆 of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
⊢ 𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵))    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘𝐴) ∈ (𝑆‘𝐵))))
Theorem	sategoelfv 35655	Condition of a valuation 𝑆 of a simplified satisfaction predicate for a Godel-set of membership: The sets in model 𝑀 corresponding to the variables 𝐴 and 𝐵 under the assignment of 𝑆 are in a membership relation in 𝑀. (Contributed by AV, 5-Nov-2023.)
⊢ 𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵))    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑆 ∈ 𝐸) → (𝑆‘𝐴) ∈ (𝑆‘𝐵))
Theorem	ex-sategoelel 35656*	Example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
⊢ 𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵))    &   ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))    ⇒   ⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑆 ∈ 𝐸)
Theorem	ex-sategoel 35657*	Instance of sategoelfv 35655 for the example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
⊢ 𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵))    &   ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))    ⇒   ⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆‘𝐴) ∈ (𝑆‘𝐵))
Theorem	satfv1fvfmla1 35658*	The value of the satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.)
⊢ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))    ⇒   ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat 𝐸)‘1o)‘𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))})
Theorem	2goelgoanfmla1 35659	Two Godel-sets of membership combined with a Godel-set for NAND is a Godel formula of height 1. (Contributed by AV, 17-Nov-2023.)
⊢ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))    ⇒   ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
Theorem	satefvfmla1 35660*	The simplified satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.)
⊢ 𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 Sat∈ 𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾) ∈ (𝑎‘𝐿))})
Theorem	ex-sategoelelomsuc 35661*	Example of a valuation of a simplified satisfaction predicate over the ordinal numbers as model for a Godel-set of membership using the properties of a successor: (𝑆‘2o) = 𝑍 ∈ suc 𝑍 = (𝑆‘2o). Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: (2o∈𝑔1o) should not be confused with 2o ∈ 1o, which is false. (Contributed by AV, 19-Nov-2023.)
⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍))    ⇒   ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)))
Theorem	ex-sategoelel12 35662	Example of a valuation of a simplified satisfaction predicate over a proper pair (of ordinal numbers) as model for a Godel-set of membership using the properties of a successor: (𝑆‘2o) = 1o ∈ 2o = (𝑆‘2o). Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: (2o∈𝑔1o) should not be confused with 2o ∈ 1o, which is false. (Contributed by AV, 19-Nov-2023.)
⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 1o, 2o))    ⇒   ⊢ 𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o))
Theorem	prv 35663	The "proves" relation on a set. A wff encoded as 𝑈 is true in a model 𝑀 iff for every valuation 𝑠 ∈ (𝑀 ↑m ω), the interpretation of the wff using the membership relation on 𝑀 is true. (Contributed by AV, 5-Nov-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀⊧𝑈 ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))
Theorem	elnanelprv 35664	The wff (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) encoded as ((𝐴∈𝑔𝐵) ⊼𝑔(𝐵∈𝑔𝐴)) is true in any model 𝑀. This is the model theoretic proof of elnanel 9526. (Contributed by AV, 5-Nov-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝑀⊧((𝐴∈𝑔𝐵)⊼𝑔(𝐵∈𝑔𝐴)))
Theorem	prv0 35665	Every wff encoded as 𝑈 is true in an "empty model" (𝑀 = ∅). Since ⊧ is defined in terms of the interpretations making the given formula true, it is not defined on the "empty model", since there are no interpretations. In particular, the empty set on the LHS of ⊧ should not be interpreted as the empty model, because ∃𝑥𝑥 = 𝑥 is not satisfied on the empty model. (Contributed by AV, 19-Nov-2023.)
⊢ (𝑈 ∈ (Fmla‘ω) → ∅⊧𝑈)
Theorem	prv1n 35666	No wff encoded as a Godel-set of membership is true in a model with only one element. (Contributed by AV, 19-Nov-2023.)
⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ {𝑋}⊧(𝐼∈𝑔𝐽))
21.6.11  Godel-sets of formulas - part 2
Syntax	cgon 35667	The Godel-set of negation. (Note that this is not a wff.)
class ¬𝑔𝑈
Syntax	cgoa 35668	The Godel-set of conjunction.
class ∧𝑔
Syntax	cgoi 35669	The Godel-set of implication.
class →𝑔
Syntax	cgoo 35670	The Godel-set of disjunction.
class ∨𝑔
Syntax	cgob 35671	The Godel-set of equivalence.
class ↔𝑔
Syntax	cgoq 35672	The Godel-set of equality.
class =𝑔
Syntax	cgox 35673	The Godel-set of existential quantification. (Note that this is not a wff.)
class ∃𝑔𝑁𝑈
Definition	df-gonot 35674	Define the Godel-set of negation. Here the argument 𝑈 is also a Godel-set corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ¬𝑔𝑈 = (𝑈⊼𝑔𝑈)
Definition	df-goan 35675*	Define the Godel-set of conjunction. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ∧𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ¬𝑔(𝑢⊼𝑔𝑣))
Definition	df-goim 35676*	Define the Godel-set of implication. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ →𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑢⊼𝑔¬𝑔𝑣))
Definition	df-goor 35677*	Define the Godel-set of disjunction. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ∨𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (¬𝑔𝑢 →𝑔 𝑣))
Definition	df-gobi 35678*	Define the Godel-set of equivalence. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ↔𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑢 →𝑔 𝑣)∧𝑔(𝑣 →𝑔 𝑢)))
Definition	df-goeq 35679*	Define the Godel-set of equality. Here the arguments 𝑥 = ⟨𝑁, 𝑃⟩ correspond to vN and vP , so (∅=𝑔1o) actually means v0 = v1 , not 0 = 1. Here we use the trick mentioned in ax-ext 2712 to introduce equality as a defined notion in terms of ∈𝑔. The expression suc (𝑢 ∪ 𝑣) = max (𝑢, 𝑣) + 1 here is a convenient way of getting a dummy variable distinct from 𝑢 and 𝑣. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ =𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ ⦋suc (𝑢 ∪ 𝑣) / 𝑤⦌∀𝑔𝑤((𝑤∈𝑔𝑢) ↔𝑔 (𝑤∈𝑔𝑣)))
Definition	df-goex 35680	Define the Godel-set of existential quantification. Here 𝑁 ∈ ω corresponds to vN , and 𝑈 represents another formula, and this expression is [∃𝑥𝜑] = ∃𝑔𝑁𝑈 where 𝑥 is the 𝑁-th variable, 𝑈 = [𝜑] is the code for 𝜑. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ∃𝑔𝑁𝑈 = ¬𝑔∀𝑔𝑁¬𝑔𝑈
21.6.12  Models of ZF
Syntax	cgze 35681	The Axiom of Extensionality.
class AxExt
Syntax	cgzr 35682	The Axiom Scheme of Replacement.
class AxRep
Syntax	cgzp 35683	The Axiom of Power Sets.
class AxPow
Syntax	cgzu 35684	The Axiom of Unions.
class AxUn
Syntax	cgzg 35685	The Axiom of Regularity.
class AxReg
Syntax	cgzi 35686	The Axiom of Infinity.
class AxInf
Syntax	cgzf 35687	The set of models of ZF.
class ZF
Definition	df-gzext 35688	The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxExt = (∀𝑔2o((2o∈𝑔∅) ↔𝑔 (2o∈𝑔1o)) →𝑔 (∅=𝑔1o))
Definition	df-gzrep 35689	The Godel-set version of the Axiom Scheme of Replacement. Since this is a scheme and not a single axiom, it manifests as a function on wffs, each giving rise to a different axiom. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxRep = (𝑢 ∈ (Fmla‘ω) ↦ (∀𝑔3o∃𝑔1o∀𝑔2o(∀𝑔1o𝑢 →𝑔 (2o=𝑔1o)) →𝑔 ∀𝑔1o∀𝑔2o((2o∈𝑔1o) ↔𝑔 ∃𝑔3o((3o∈𝑔∅)∧𝑔∀𝑔1o𝑢))))
Definition	df-gzpow 35690	The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxPow = ∃𝑔1o∀𝑔2o(∀𝑔1o((1o∈𝑔2o) ↔𝑔 (1o∈𝑔∅)) →𝑔 (2o∈𝑔1o))
Definition	df-gzun 35691	The Godel-set version of the Axiom of Unions. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxUn = ∃𝑔1o∀𝑔2o(∃𝑔1o((2o∈𝑔1o)∧𝑔(1o∈𝑔∅)) →𝑔 (2o∈𝑔1o))
Definition	df-gzreg 35692	The Godel-set version of the Axiom of Regularity. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxReg = (∃𝑔1o(1o∈𝑔∅) →𝑔 ∃𝑔1o((1o∈𝑔∅)∧𝑔∀𝑔2o((2o∈𝑔1o) →𝑔 ¬𝑔(2o∈𝑔∅))))
Definition	df-gzinf 35693	The Godel-set version of the Axiom of Infinity. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxInf = ∃𝑔1o((∅∈𝑔1o)∧𝑔∀𝑔2o((2o∈𝑔1o) →𝑔 ∃𝑔∅((2o∈𝑔∅)∧𝑔(∅∈𝑔1o))))
Definition	df-gzf 35694*	Define the class of all (transitive) models of ZF. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ZF = {𝑚 ∣ ((Tr 𝑚 ∧ 𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈ (Fmla‘ω)𝑚⊧(AxRep‘𝑢))}
21.6.13  Metamath formal systems This is a formalization of Appendix C of the Metamath book, which describes the mathematical representation of a formal system, of which set.mm (this file) is one.
Syntax	cmcn 35695	The set of constants.
class mCN
Syntax	cmvar 35696	The set of variables.
class mVR
Syntax	cmty 35697	The type function.
class mType
Syntax	cmvt 35698	The set of variable typecodes.
class mVT
Syntax	cmtc 35699	The set of typecodes.
class mTC
Syntax	cmax 35700	The set of axioms.
class mAx