HomeTheorem List (p. 363 of 507)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | axextprim 36201 | ax-ext 2734 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
| ⊢ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧)) | ||
| Theorem | axrepprim 36202 | ax-rep 5237 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
| ⊢ ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧 ∈ 𝑥 → ¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧 ∈ 𝑥))) | ||
| Theorem | axunprim 36203 | ax-un 7734 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
| ⊢ ¬ ∀𝑥 ¬ ∀𝑦(¬ ∀𝑥(𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
| Theorem | axpowprim 36204 | ax-pow 5335 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
| ⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) → 𝑥 = 𝑦) | ||
| Theorem | axregprim 36205 | ax-reg 9552 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
| ⊢ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) | ||
| Theorem | axinfprim 36206 | ax-inf 9605 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.) |
| ⊢ ¬ ∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ (𝑦 ∈ 𝑥 → ¬ ∀𝑦(𝑦 ∈ 𝑥 → ¬ ∀𝑧(𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥)))) | ||
| Theorem | axacprim 36207 | ax-ac 10449 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.) |
| ⊢ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)))))) | ||
| Theorem | untelirr 36208* | We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 36290). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.) |
| ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴) | ||
| Theorem | untuni 36209* | The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.) |
| ⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥) | ||
| Theorem | untsucf 36210* | If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.) |
| ⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦) | ||
| Theorem | unt0 36211 | The empty set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
| ⊢ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ 𝑥 | ||
| Theorem | untint 36212* | If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.) |
| ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦) | ||
| Theorem | efrunt 36213* | If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.) |
| ⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥) | ||
| Theorem | untangtr 36214* | A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.) |
| ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)) | ||
| Theorem | 3jaodd 36215 | Double deduction form of 3jaoi 1453. (Contributed by Scott Fenton, 20-Apr-2011.) |
| ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) & ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂))) & ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃 ∨ 𝜏) → 𝜂))) | ||
| Theorem | 3orit 36216 | Closed form of 3ori 1450. (Contributed by Scott Fenton, 20-Apr-2011.) |
| ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)) | ||
| Theorem | biimpexp 36217 | A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.) |
| ⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜒))) | ||
| Theorem | nepss 36218 | Two classes are unequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.) |
| ⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵)) | ||
| Theorem | 3ccased 36219 | Triple disjunction form of ccased 1053. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ (𝜑 → ((𝜒 ∧ 𝜂) → 𝜓)) & ⊢ (𝜑 → ((𝜒 ∧ 𝜁) → 𝜓)) & ⊢ (𝜑 → ((𝜒 ∧ 𝜎) → 𝜓)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜂) → 𝜓)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜁) → 𝜓)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜎) → 𝜓)) & ⊢ (𝜑 → ((𝜏 ∧ 𝜂) → 𝜓)) & ⊢ (𝜑 → ((𝜏 ∧ 𝜁) → 𝜓)) & ⊢ (𝜑 → ((𝜏 ∧ 𝜎) → 𝜓)) ⇒ ⊢ (𝜑 → (((𝜒 ∨ 𝜃 ∨ 𝜏) ∧ (𝜂 ∨ 𝜁 ∨ 𝜎)) → 𝜓)) | ||
| Theorem | dfso3 36220* | Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.) |
| ⊢ (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | ||
| Theorem | brtpid1 36221 | A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.) |
| ⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵 | ||
| Theorem | brtpid2 36222 | A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.) |
| ⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵 | ||
| Theorem | brtpid3 36223 | A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.) |
| ⊢ 𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵 | ||
| Theorem | iota5f 36224* | A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) | ||
| Theorem | jath 36225 | Closed form of ja 188. Proved using the completeness script. (Proof modification is discouraged.) (Contributed by Scott Fenton, 13-Dec-2021.) |
| ⊢ ((¬ 𝜑 → 𝜒) → ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → 𝜒))) | ||
| Theorem | xpab 36226* | Cartesian product of two class abstractions. (Contributed by Scott Fenton, 19-Aug-2024.) |
| ⊢ ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)} | ||
| Theorem | nnuni 36227 | The union of a finite ordinal is a finite ordinal. (Contributed by Scott Fenton, 17-Oct-2024.) |
| ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) | ||
| Theorem | sqdivzi 36228 | Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) | ||
| Theorem | supfz 36229 | The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) |
| ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁) | ||
| Theorem | inffz 36230 | The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by AV, 10-Oct-2021.) |
| ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀) | ||
| Theorem | fz0n 36231 | The sequence (0...(𝑁 − 1)) is empty iff 𝑁 is zero. (Contributed by Scott Fenton, 16-May-2014.) |
| ⊢ (𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0)) | ||
| Theorem | shftvalg 36232 | Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) | ||
| Theorem | divcnvlin 36233* | Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((𝑘 + 𝐴) / (𝑘 + 𝐵))) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 1) | ||
| Theorem | climlec3 36234* | Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
| Theorem | iexpire 36235 | i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.) |
| ⊢ (i↑𝑐i) ∈ ℝ | ||
| Theorem | bcneg1 36236 | The binomial coefficient over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.) |
| ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) | ||
| Theorem | bcm1nt 36237 | The proportion of one binomial coefficient to another with 𝑁 decreased by 1. (Contributed by Scott Fenton, 23-Jun-2020.) |
| ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (𝑁C𝐾) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁 − 𝐾)))) | ||
| Theorem | bcprod 36238* | A product identity for binomial coefficients. (Contributed by Scott Fenton, 23-Jun-2020.) |
| ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁))) | ||
| Theorem | bccolsum 36239* | A column-sum rule for binomial coefficients. (Contributed by Scott Fenton, 24-Jun-2020.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))) | ||
| Theorem | iprodefisumlem 36240 | Lemma for iprodefisum 36241. (Contributed by Scott Fenton, 11-Feb-2018.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) ⇒ ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹))) | ||
| Theorem | iprodefisum 36241* | Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵)) | ||
| Theorem | iprodgam 36242* | An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.) |
| ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) ⇒ ⊢ (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴)) | ||
| Theorem | faclimlem1 36243* | Lemma for faclim 36246. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.) |
| ⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) = (𝑥 ∈ ℕ ↦ ((𝑀 + 1) · ((𝑥 + 1) / (𝑥 + (𝑀 + 1)))))) | ||
| Theorem | faclimlem2 36244* | Lemma for faclim 36246. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.) |
| ⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) ⇝ (𝑀 + 1)) | ||
| Theorem | faclimlem3 36245 | Lemma for faclim 36246. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.) |
| ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (((1 + (1 / 𝐵))↑(𝑀 + 1)) / (1 + ((𝑀 + 1) / 𝐵))) = ((((1 + (1 / 𝐵))↑𝑀) / (1 + (𝑀 / 𝐵))) · (((1 + (𝑀 / 𝐵)) · (1 + (1 / 𝐵))) / (1 + ((𝑀 + 1) / 𝐵))))) | ||
| Theorem | faclim 36246* | An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))) ⇒ ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴)) | ||
| Theorem | iprodfac 36247* | An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.) |
| ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘)))) | ||
| Theorem | faclim2 36248* | Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀)))) ⇒ ⊢ (𝑀 ∈ ℕ0 → 𝐹 ⇝ 1) | ||
| Theorem | gcd32 36249 | Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = ((𝐴 gcd 𝐶) gcd 𝐵)) | ||
| Theorem | gcdabsorb 36250 | Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐵) = (𝐴 gcd 𝐵)) | ||
| Theorem | dftr6 36251 | A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) | ||
| Theorem | coep 36252* | Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥) | ||
| Theorem | coepr 36253* | Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵) | ||
| Theorem | dffr5 36254 | A quantifier-free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.) |
| ⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅))) | ||
| Theorem | dfso2 36255 | Quantifier-free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.) |
| ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)))) | ||
| Theorem | br8 36256* | Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
| ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎)) & ⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)} ⇒ ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜌)) | ||
| Theorem | br6 36257* | Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
| ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ 𝜑)} ⇒ ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄) ∧ (𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄)) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩𝑅⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ 𝜁)) | ||
| Theorem | br4 36258* | Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.) |
| ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)} ⇒ ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜏)) | ||
| Theorem | cnvco1 36259 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
| ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) | ||
| Theorem | cnvco2 36260 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
| ⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴) | ||
| Theorem | eldm3 36261 | Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.) |
| ⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅) | ||
| Theorem | elrn3 36262 | Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.) |
| ⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) | ||
| Theorem | pocnv 36263 | The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
| ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴) | ||
| Theorem | socnv 36264 | The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
| ⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴) | ||
| Theorem | elintfv 36265* | Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.) |
| ⊢ 𝑋 ∈ V ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))) | ||
| Theorem | funpsstri 36266 | A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.) |
| ⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹)) | ||
| Theorem | fundmpss 36267 | If a class 𝐹 is a proper subset of a function 𝐺, then dom 𝐹 ⊊ dom 𝐺. (Contributed by Scott Fenton, 20-Apr-2011.) |
| ⊢ (Fun 𝐺 → (𝐹 ⊊ 𝐺 → dom 𝐹 ⊊ dom 𝐺)) | ||
| Theorem | funsseq 36268 | Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
| ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺 ↔ 𝐹 ⊆ 𝐺)) | ||
| Theorem | fununiq 36269 | The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)) | ||
| Theorem | funbreq 36270 | An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶)) | ||
| Theorem | br1steq 36271 | Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴) | ||
| Theorem | br2ndeq 36272 | Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵) | ||
| Theorem | dfdm5 36273 | Definition of domain in terms of 1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
| ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴) | ||
| Theorem | dfrn5 36274 | Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
| ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴) | ||
| Theorem | opelco3 36275 | Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.) |
| ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))) | ||
| Theorem | elima4 36276 | Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.) |
| ⊢ (𝐴 ∈ (𝑅 “ 𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅) | ||
| Theorem | fv1stcnv 36277 | The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.) |
| ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩) | ||
| Theorem | fv2ndcnv 36278 | The value of the converse of 2nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.) |
| ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩) | ||
| Theorem | elpotr 36279* | A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.) |
| ⊢ (∀𝑧 ∈ 𝐴 Tr 𝑧 → E Po 𝐴) | ||
| Theorem | dford5reg 36280 | Given ax-reg 9552, an ordinal is a transitive class totally ordered by the membership relation. (Contributed by Scott Fenton, 28-Jan-2011.) |
| ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴)) | ||
| Theorem | dfon2lem1 36281 | Lemma for dfon2 36290. (Contributed by Scott Fenton, 28-Feb-2011.) |
| ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} | ||
| Theorem | dfon2lem2 36282* | Lemma for dfon2 36290. (Contributed by Scott Fenton, 28-Feb-2011.) |
| ⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴 | ||
| Theorem | dfon2lem3 36283* | Lemma for dfon2 36290. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.) |
| ⊢ (𝐴 ∈ 𝑉 → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧))) | ||
| Theorem | dfon2lem4 36284* | Lemma for dfon2 36290. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | ||
| Theorem | dfon2lem5 36285* | Lemma for dfon2 36290. Two sets satisfying the new definition also satisfy trichotomy with respect to ∈. (Contributed by Scott Fenton, 25-Feb-2011.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | ||
| Theorem | dfon2lem6 36286* | Lemma for dfon2 36290. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.) |
| ⊢ ((Tr 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑧((𝑧 ⊊ 𝑥 ∧ Tr 𝑧) → 𝑧 ∈ 𝑥)) → ∀𝑦((𝑦 ⊊ 𝑆 ∧ Tr 𝑦) → 𝑦 ∈ 𝑆)) | ||
| Theorem | dfon2lem7 36287* | Lemma for dfon2 36290. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵))) | ||
| Theorem | dfon2lem8 36288* | Lemma for dfon2 36290. The intersection of a nonempty class 𝐴 of new ordinals is itself a new ordinal and is contained within 𝐴 (Contributed by Scott Fenton, 26-Feb-2011.) |
| ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)) → (∀𝑧((𝑧 ⊊ ∩ 𝐴 ∧ Tr 𝑧) → 𝑧 ∈ ∩ 𝐴) ∧ ∩ 𝐴 ∈ 𝐴)) | ||
| Theorem | dfon2lem9 36289* | Lemma for dfon2 36290. A class of new ordinals is well-founded by E. (Contributed by Scott Fenton, 3-Mar-2011.) |
| ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → E Fr 𝐴) | ||
| Theorem | dfon2 36290* | On consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers", American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.) |
| ⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)} | ||
| Theorem | rdgprc0 36291 | The value of the recursive definition generator at ∅ when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅) | ||
| Theorem | rdgprc 36292 | The value of the recursive definition generator when 𝐼 is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅)) | ||
| Theorem | dfrdg2 36293* | Alternate definition of the recursive function generator when 𝐼 is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ (𝐼 ∈ 𝑉 → rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}) | ||
| Theorem | dfrdg3 36294* | Generalization of dfrdg2 36293 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))} | ||
| Theorem | axextdfeq 36295 | A version of ax-ext 2734 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.) |
| ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))) | ||
| Theorem | ax8dfeq 36296 | A version of ax-8 2144 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.) |
| ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦)) | ||
| Theorem | axextdist 36297 | ax-ext 2734 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)) | ||
| Theorem | axextbdist 36298 | axextb 2737 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))) | ||
| Theorem | 19.12b 36299* | Version of 19.12vv 2378 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) | ||
| Theorem | exnel 36300 | There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦 | ||
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