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Type | Label | Description |
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Statement | ||
Theorem | satfdmfmla 33501 | 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 33502 | Lemma for satffunlem1lem1 33503 and satffunlem2lem1 33505. (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 33503* | Lemma for satffunlem1 33508. (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 33504* | Lemma 2 for satffunlem1 33508. (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 33505* | Lemma 1 for satffunlem2 33509. (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 33506* | 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 33507* | Lemma 2 for satffunlem2 33509. (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 33508 | Lemma 1 for satffun 33510: induction basis. (Contributed by AV, 28-Oct-2023.) |
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅)) | ||
Theorem | satffunlem2 33509 | Lemma 2 for satffun 33510: 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 33510 | 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 33511 | 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 33512 | 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 33513 | 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 33514* | 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 33515 | 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 33516 | 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 33517 | 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 33518 | 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 33519* | 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 33520 | 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 33521 | 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 33522* | 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 33523* | Instance of sategoelfv 33521 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 33524* | 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 33525 | 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 33526* | 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 33527* | 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 33528 | 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 33529 | 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 33530 | 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 9443. (Contributed by AV, 5-Nov-2023.) |
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝑀⊧((𝐴∈𝑔𝐵)⊼𝑔(𝐵∈𝑔𝐴))) | ||
Theorem | prv0 33531 | 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 33532 | 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)) | ||
Syntax | cgon 33533 | The Godel-set of negation. (Note that this is not a wff.) |
class ¬𝑔𝑈 | ||
Syntax | cgoa 33534 | The Godel-set of conjunction. |
class ∧𝑔 | ||
Syntax | cgoi 33535 | The Godel-set of implication. |
class →𝑔 | ||
Syntax | cgoo 33536 | The Godel-set of disjunction. |
class ∨𝑔 | ||
Syntax | cgob 33537 | The Godel-set of equivalence. |
class ↔𝑔 | ||
Syntax | cgoq 33538 | The Godel-set of equality. |
class =𝑔 | ||
Syntax | cgox 33539 | The Godel-set of existential quantification. (Note that this is not a wff.) |
class ∃𝑔𝑁𝑈 | ||
Definition | df-gonot 33540 | 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 33541* | 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 33542* | 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 33543* | 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 33544* | 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 33545* | 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 2708 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 33546 | 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 | ||
Syntax | cgze 33547 | The Axiom of Extensionality. |
class AxExt | ||
Syntax | cgzr 33548 | The Axiom Scheme of Replacement. |
class AxRep | ||
Syntax | cgzp 33549 | The Axiom of Power Sets. |
class AxPow | ||
Syntax | cgzu 33550 | The Axiom of Unions. |
class AxUn | ||
Syntax | cgzg 33551 | The Axiom of Regularity. |
class AxReg | ||
Syntax | cgzi 33552 | The Axiom of Infinity. |
class AxInf | ||
Syntax | cgzf 33553 | The set of models of ZF. |
class ZF | ||
Definition | df-gzext 33554 | 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 33555 | 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 33556 | 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 33557 | 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 33558 | 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 33559 | 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 33560* | 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))} | ||
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 33561 | The set of constants. |
class mCN | ||
Syntax | cmvar 33562 | The set of variables. |
class mVR | ||
Syntax | cmty 33563 | The type function. |
class mType | ||
Syntax | cmvt 33564 | The set of variable typecodes. |
class mVT | ||
Syntax | cmtc 33565 | The set of typecodes. |
class mTC | ||
Syntax | cmax 33566 | The set of axioms. |
class mAx | ||
Syntax | cmrex 33567 | The set of raw expressions. |
class mREx | ||
Syntax | cmex 33568 | The set of expressions. |
class mEx | ||
Syntax | cmdv 33569 | The set of distinct variables. |
class mDV | ||
Syntax | cmvrs 33570 | The variables in an expression. |
class mVars | ||
Syntax | cmrsub 33571 | The set of raw substitutions. |
class mRSubst | ||
Syntax | cmsub 33572 | The set of substitutions. |
class mSubst | ||
Syntax | cmvh 33573 | The set of variable hypotheses. |
class mVH | ||
Syntax | cmpst 33574 | The set of pre-statements. |
class mPreSt | ||
Syntax | cmsr 33575 | The reduct of a pre-statement. |
class mStRed | ||
Syntax | cmsta 33576 | The set of statements. |
class mStat | ||
Syntax | cmfs 33577 | The set of formal systems. |
class mFS | ||
Syntax | cmcls 33578 | The closure of a set of statements. |
class mCls | ||
Syntax | cmpps 33579 | The set of provable pre-statements. |
class mPPSt | ||
Syntax | cmthm 33580 | The set of theorems. |
class mThm | ||
Definition | df-mcn 33581 | Define the set of constants in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mCN = Slot 1 | ||
Definition | df-mvar 33582 | Define the set of variables in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVR = Slot 2 | ||
Definition | df-mty 33583 | Define the type function in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mType = Slot 3 | ||
Definition | df-mtc 33584 | Define the set of typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mTC = Slot 4 | ||
Definition | df-mmax 33585 | Define the set of axioms in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mAx = Slot 5 | ||
Definition | df-mvt 33586 | Define the set of variable typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡)) | ||
Definition | df-mrex 33587 | Define the set of "raw expressions", which are expressions without a typecode attached. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡))) | ||
Definition | df-mex 33588 | Define the set of expressions, which are strings of constants and variables headed by a typecode constant. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡))) | ||
Definition | df-mdv 33589 | Define the set of distinct variable conditions, which are pairs of distinct variables. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I )) | ||
Definition | df-mvrs 33590* | Define the set of variables in an expression. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)))) | ||
Definition | df-mrsub 33591* | Define a substitution of raw expressions given a mapping from variables to expressions. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mRSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))))) | ||
Definition | df-msub 33592* | Define a substitution of expressions given a mapping from variables to expressions. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))〉))) | ||
Definition | df-mvh 33593* | Define the mapping from variables to their variable hypothesis. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉)) | ||
Definition | df-mpst 33594* | Define the set of all pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡))) | ||
Definition | df-msr 33595* | Define the reduct of a pre-statement. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ ⦋(2nd ‘(1st ‘𝑠)) / ℎ⦌⦋(2nd ‘𝑠) / 𝑎⦌〈((1st ‘(1st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎〉)) | ||
Definition | df-msta 33596 | Define the set of all statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡)) | ||
Definition | df-mfs 33597* | Define the set of all formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mFS = {𝑡 ∣ ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ (◡(mType‘𝑡) “ {𝑣}) ∈ Fin))} | ||
Definition | df-mcls 33598* | Define the closure of a set of statements relative to a set of disjointness constraints. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mCls = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))})) | ||
Definition | df-mpps 33599* | Define the set of provable pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mPPSt = (𝑡 ∈ V ↦ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))}) | ||
Definition | df-mthm 33600 | Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡)))) |
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