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Theorem List for Metamath Proof Explorer - 32501-32600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsatom 32501* The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at omega (ω). (Contributed by AV, 6-Oct-2023.)
((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘ω) = 𝑛 ∈ ω ((𝑀 Sat 𝐸)‘𝑛))
 
Theoremsatfvsucom 32502* The satisfaction predicate as function over wff codes at a successor of ω. (Contributed by AV, 22-Sep-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊𝑁 ∈ suc ω) → (𝑆𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑓 (∃𝑣𝑓 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})‘𝑁))
 
Theoremsatfv0 32503* The value of the satisfaction predicate as function over wff codes at . (Contributed by AV, 8-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})
 
Theoremsatfvsuclem1 32504* Lemma 1 for satfvsuc 32506. (Contributed by AV, 8-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀m ω))} ∈ V)
 
Theoremsatfvsuclem2 32505* Lemma 2 for satfvsuc 32506. (Contributed by AV, 8-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} ∈ V)
 
Theoremsatfvsuc 32506* The value of the satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 10-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (𝑆‘suc 𝑁) = ((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}))
 
Theoremsatfv1lem 32507* Lemma for satfv1 32508. (Contributed by AV, 9-Nov-2023.)
((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀m ω) ∣ (𝑏𝐼)𝐸(𝑏𝐽)}} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))})
 
Theoremsatfv1 32508* The value of the satisfaction predicate as function over wff codes of height 1. (Contributed by AV, 9-Nov-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊) → (𝑆‘1o) = ((𝑆‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))}))
 
Theoremsatfsschain 32509 The binary relation of a satisfaction predicate as function over wff codes is an increasing chain (with respect to inclusion). (Contributed by AV, 15-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)       (((𝑀𝑉𝐸𝑊) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵𝐴 → (𝑆𝐵) ⊆ (𝑆𝐴)))
 
Theoremsatfvsucsuc 32510* The satisfaction predicate as function over wff codes of height (𝑁 + 1), expressed by the minimally necessary satisfaction predicates as function over wff codes of height 𝑁. (Contributed by AV, 21-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝐴 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))    &   𝐵 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}       ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (𝑆‘suc suc 𝑁) = ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))}))
 
Theoremsatfbrsuc 32511* 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 = (𝑆𝑁)       (((𝑀𝑉𝐸𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴𝑋𝐵𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢𝑃 (∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))))
 
Theoremsatfrel 32512 The value of the satisfaction predicate as function over wff codes at a natural number is a relation. (Contributed by AV, 12-Oct-2023.)
((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))
 
Theoremsatfdmlem 32513* Lemma for satfdm 32514. (Contributed by AV, 12-Oct-2023.)
(((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))))
 
Theoremsatfdm 32514* 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.)
(((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))
 
Theoremsatfrnmapom 32515 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.)
((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀m ω))
 
Theoremsatfv0fun 32516 The value of the satisfaction predicate as function over wff codes at is a function. (Contributed by AV, 15-Oct-2023.)
((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
 
Theoremsatf0 32517* 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 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}) ↾ suc ω)
 
Theoremsatf0sucom 32518* 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 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑁))
 
Theoremsatf00 32519* 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 = (𝑖𝑔𝑗))}
 
Theoremsatf0suclem 32520* Lemma for satf0suc 32521, sat1el2xp 32524 and fmlasuc0 32529. (Contributed by AV, 19-Sep-2023.)
((𝑋𝑈𝑌𝑉𝑍𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶))} ∈ V)
 
Theoremsatf0suc 32521* 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) = ((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
 
Theoremsatf0op 32522* 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 ∈ (𝑆𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
 
Theoremsatf0n0 32523 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))
 
Theoremsat1el2xp 32524* 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)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
 
Theoremfmlafv 32525 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))
 
Theoremfmla 32526 The set of all valid Godel formulas. (Contributed by AV, 20-Sep-2023.)
(Fmla‘ω) = 𝑛 ∈ ω (Fmla‘𝑛)
 
Theoremfmla0 32527* 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 = (𝑖𝑔𝑗)}
 
Theoremfmla0xp 32528 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))
 
Theoremfmlasuc0 32529* 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))}))
 
Theoremfmlafvel 32530 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)))
 
Theoremfmlasuc 32531* 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 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
 
Theoremfmla1 32532* The valid Godel formulas of height 1 is the set of all formulas of the form (𝑎𝑔𝑏) and 𝑔𝑘𝑎 with atoms 𝑎, 𝑏 of the form 𝑥𝑦. (Contributed by AV, 20-Sep-2023.)
(Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))})
 
Theoremisfmlasuc 32533* The characterization of a Godel formula of height at least 1. (Contributed by AV, 14-Oct-2023.)
((𝑁 ∈ ω ∧ 𝐹𝑉) → (𝐹 ∈ (Fmla‘suc 𝑁) ↔ (𝐹 ∈ (Fmla‘𝑁) ∨ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))))
 
Theoremfmlasssuc 32534 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))
 
Theoremfmlaomn0 32535 The empty set is not a Godel formula of any height. (Contributed by AV, 21-Oct-2023.)
(𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))
 
Theoremfmlan0 32536 The empty set is not a Godel formula. (Contributed by AV, 19-Nov-2023.)
∅ ∉ (Fmla‘ω)
 
Theoremgonan0 32537 The "Godel-set of NAND" is a Godel formula of at least height 1. (Contributed by AV, 21-Oct-2023.)
((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
 
Theoremgoaln0 32538* 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 ≠ ∅)
 
Theoremgonarlem 32539* Lemma for gonar 32540 (induction step). (Contributed by AV, 21-Oct-2023.)
(𝑁 ∈ ω → (((𝑎𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁))) → ((𝑎𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
 
Theoremgonar 32540* 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 ∧ (𝑎𝑔𝑏) ∈ (Fmla‘𝑁)) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁)))
 
Theoremgoalrlem 32541* Lemma for goalr 32542 (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))))
 
Theoremgoalr 32542* 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))
 
Theoremfmla0disjsuc 32543* 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 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅
 
Theoremfmlasucdisj 32544* 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 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ∨ ∃𝑢 ∈ (Fmla‘𝑁)∃𝑣 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑢𝑔𝑣))}) = ∅)
 
Theoremsatfdmfmla 32545 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.)
((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))
 
Theoremsatffunlem 32546 Lemma for satffunlem1lem1 32547 and satffunlem2lem1 32549. (Contributed by AV, 27-Oct-2023.)
(((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) ∧ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))) → 𝑦 = 𝑤)
 
Theoremsatffunlem1lem1 32547* Lemma for satffunlem1 32552. (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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})
 
Theoremsatffunlem1lem2 32548* Lemma 2 for satffunlem1 32552. (Contributed by AV, 23-Oct-2023.)
((𝑀𝑉𝐸𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}) = ∅)
 
Theoremsatffunlem2lem1 32549* Lemma 1 for satffunlem2 32553. (Contributed by AV, 28-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝐴 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))    &   𝐵 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}       ((Fun (𝑆‘suc 𝑁) ∧ (𝑆𝑁) ⊆ (𝑆‘suc 𝑁)) → Fun {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))})
 
Theoremdmopab3rexdif 32550* The domain of an ordered pair class abstraction with three nested restricted existential quantifiers with differences. (Contributed by AV, 25-Oct-2023.)
((∀𝑢𝑈 (∀𝑣𝑈 𝐵𝑋 ∧ ∀𝑖𝐼 𝐷𝑊) ∧ 𝑆𝑈) → dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑈𝑆)(∃𝑣𝑈 (𝑥 = 𝐴𝑦 = 𝐵) ∨ ∃𝑖𝐼 (𝑥 = 𝐶𝑦 = 𝐷)) ∨ ∃𝑢𝑆𝑣 ∈ (𝑈𝑆)(𝑥 = 𝐴𝑦 = 𝐵))} = {𝑥 ∣ (∃𝑢 ∈ (𝑈𝑆)(∃𝑣𝑈 𝑥 = 𝐴 ∨ ∃𝑖𝐼 𝑥 = 𝐶) ∨ ∃𝑢𝑆𝑣 ∈ (𝑈𝑆)𝑥 = 𝐴)})
 
Theoremsatffunlem2lem2 32551* Lemma 2 for satffunlem2 32553. (Contributed by AV, 27-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝐴 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))    &   𝐵 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}       (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (dom (𝑆‘suc 𝑁) ∩ dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))}) = ∅)
 
Theoremsatffunlem1 32552 Lemma 1 for satffun 32554: induction basis. (Contributed by AV, 28-Oct-2023.)
((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))
 
Theoremsatffunlem2 32553 Lemma 2 for satffun 32554: induction step. (Contributed by AV, 28-Oct-2023.)
((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁)))
 
Theoremsatffun 32554 The value of the satisfaction predicate as function over wff codes at a natural number is a function. (Contributed by AV, 28-Oct-2023.)
((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁))
 
Theoremsatff 32555 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.)
((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑁):(Fmla‘𝑁)⟶𝒫 (𝑀m ω))
 
Theoremsatfun 32556 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.)
((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))
 
Theoremsatfvel 32557 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.)
(((𝑀𝑉𝐸𝑊) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈)) → 𝑆:ω⟶𝑀)
 
Theoremsatfv0fvfmla0 32558* The value of the satisfaction predicate as function over a wff code at . (Contributed by AV, 2-Nov-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))})
 
Theoremsatefv 32559 The simplified satisfaction predicate as function over wff codes in the model 𝑀 at the code 𝑈. (Contributed by AV, 30-Oct-2023.)
((𝑀𝑉𝑈𝑊) → (𝑀 Sat 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
 
Theoremsate0 32560 The simplified satisfaction predicate for any wff code over an empty model. (Contributed by AV, 6-Oct-2023.) (Revised by AV, 5-Nov-2023.)
(𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
 
Theoremsatef 32561 The simplified satisfaction predicate as function over wff codes over an empty model. (Contributed by AV, 30-Oct-2023.)
((𝑀𝑉𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (𝑀 Sat 𝑈)) → 𝑆:ω⟶𝑀)
 
Theoremsate0fv0 32562 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 = ∅))
 
Theoremsatefvfmla0 32563* The simplified satisfaction predicate for wff codes of height 0. (Contributed by AV, 4-Nov-2023.)
((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat 𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
 
Theoremsategoelfvb 32564 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 (𝐴𝑔𝐵))       ((𝑀𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆𝐸 ↔ (𝑆 ∈ (𝑀m ω) ∧ (𝑆𝐴) ∈ (𝑆𝐵))))
 
Theoremsategoelfv 32565 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 (𝐴𝑔𝐵))       ((𝑀𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑆𝐸) → (𝑆𝐴) ∈ (𝑆𝐵))
 
Theoremex-sategoelel 32566* Example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
𝐸 = (𝑀 Sat (𝐴𝑔𝐵))    &   𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))       (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆𝐸)
 
Theoremex-sategoel 32567* Instance of sategoelfv 32565 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 (𝐴𝑔𝐵))    &   𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))       (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐴) ∈ (𝑆𝐵))
 
Theoremsatfv1fvfmla1 32568* 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 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))       (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat 𝐸)‘1o)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))})
 
Theorem2goelgoanfmla1 32569 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 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))       (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
 
Theoremsatefvfmla1 32570* The simplified satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.)
𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))       ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 Sat 𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼) ∈ (𝑎𝐽) ∨ ¬ (𝑎𝐾) ∈ (𝑎𝐿))})
 
Theoremex-sategoelelomsuc 32571* 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: (2o𝑔1o) 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 (2o𝑔1o)))
 
Theoremex-sategoelel12 32572 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: (2o𝑔1o) 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 (2o𝑔1o))
 
Theoremprv 32573 The "proves" relation on a set. A wff encoded as 𝑈 is true in a model 𝑀 iff for every valuation 𝑠 ∈ (𝑀m ω), the interpretation of the wff using the membership relation on 𝑀 is true. (Contributed by AV, 5-Nov-2023.)
((𝑀𝑉𝑈𝑊) → (𝑀𝑈 ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))
 
Theoremelnanelprv 32574 The wff (𝐴𝐵𝐵𝐴) encoded as ((𝐴𝑔𝐵) 𝑔(𝐵𝑔𝐴)) is true in any model 𝑀. This is the model theoretic proof of elnanel 9059. (Contributed by AV, 5-Nov-2023.)
((𝑀𝑉𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝑀⊧((𝐴𝑔𝐵)⊼𝑔(𝐵𝑔𝐴)))
 
Theoremprv0 32575 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)
 
Theoremprv1n 32576 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 ∧ 𝑋𝑉) → ¬ {𝑋}⊧(𝐼𝑔𝐽))
 
20.6.11  Godel-sets of formulas - part 2
 
Syntaxcgon 32577 The Godel-set of negation. (Note that this is not a wff.)
class ¬𝑔𝑈
 
Syntaxcgoa 32578 The Godel-set of conjunction.
class 𝑔
 
Syntaxcgoi 32579 The Godel-set of implication.
class 𝑔
 
Syntaxcgoo 32580 The Godel-set of disjunction.
class 𝑔
 
Syntaxcgob 32581 The Godel-set of equivalence.
class 𝑔
 
Syntaxcgoq 32582 The Godel-set of equality.
class =𝑔
 
Syntaxcgox 32583 The Godel-set of existential quantification. (Note that this is not a wff.)
class 𝑔𝑁𝑈
 
Definitiondf-gonot 32584 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 = (𝑈𝑔𝑈)
 
Definitiondf-goan 32585* 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(𝑢𝑔𝑣))
 
Definitiondf-goim 32586* 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 ↦ (𝑢𝑔¬𝑔𝑣))
 
Definitiondf-goor 32587* 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 ↦ (¬𝑔𝑢𝑔 𝑣))
 
Definitiondf-gobi 32588* 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 ↦ ((𝑢𝑔 𝑣)∧𝑔(𝑣𝑔 𝑢)))
 
Definitiondf-goeq 32589* 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 2793 to introduce equality as a defined notion in terms of 𝑔. The expression suc (𝑢𝑣) = 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 (𝑢𝑣) / 𝑤𝑔𝑤((𝑤𝑔𝑢) ↔𝑔 (𝑤𝑔𝑣)))
 
Definitiondf-goex 32590 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 = ¬𝑔𝑔𝑁¬𝑔𝑈
 
20.6.12  Models of ZF
 
Syntaxcgze 32591 The Axiom of Extensionality.
class AxExt
 
Syntaxcgzr 32592 The Axiom Scheme of Replacement.
class AxRep
 
Syntaxcgzp 32593 The Axiom of Power Sets.
class AxPow
 
Syntaxcgzu 32594 The Axiom of Unions.
class AxUn
 
Syntaxcgzg 32595 The Axiom of Regularity.
class AxReg
 
Syntaxcgzi 32596 The Axiom of Infinity.
class AxInf
 
Syntaxcgzf 32597 The set of models of ZF.
class ZF
 
Definitiondf-gzext 32598 The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxExt = (∀𝑔2o((2o𝑔∅) ↔𝑔 (2o𝑔1o)) →𝑔 (∅=𝑔1o))
 
Definitiondf-gzrep 32599 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) ↦ (∀𝑔3o𝑔1o𝑔2o(∀𝑔1o𝑢𝑔 (2o=𝑔1o)) →𝑔𝑔1o𝑔2o((2o𝑔1o) ↔𝑔𝑔3o((3o𝑔∅)∧𝑔𝑔1o𝑢))))
 
Definitiondf-gzpow 32600 The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxPow = ∃𝑔1o𝑔2o(∀𝑔1o((1o𝑔2o) ↔𝑔 (1o𝑔∅)) →𝑔 (2o𝑔1o))
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