HomeTheorem List (p. 358 of 498)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | untangtr 35701* | 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 35702 | Double deduction form of 3jaoi 1430. (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 35703 | Closed form of 3ori 1426. (Contributed by Scott Fenton, 20-Apr-2011.) |
| ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)) | ||
| Theorem | biimpexp 35704 | 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 35705 | 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 35706 | Triple disjunction form of ccased 1038. (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 35707* | 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 35708 | A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.) |
| ⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵 | ||
| Theorem | brtpid2 35709 | A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.) |
| ⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵 | ||
| Theorem | brtpid3 35710 | A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.) |
| ⊢ 𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵 | ||
| Theorem | iota5f 35711* | 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 35712 | Closed form of ja 186. 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 35713* | Cartesian product of two class abstractions. (Contributed by Scott Fenton, 19-Aug-2024.) |
| ⊢ ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)} | ||
| Theorem | nnuni 35714 | The union of a finite ordinal is a finite ordinal. (Contributed by Scott Fenton, 17-Oct-2024.) |
| ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) | ||
| Theorem | sqdivzi 35715 | 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 35716 | 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 35717 | 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 35718 | 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 35719 | 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 35720* | 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 35721* | 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 35722 | i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.) |
| ⊢ (i↑𝑐i) ∈ ℝ | ||
| Theorem | bcneg1 35723 | The binomial coefficient over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.) |
| ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) | ||
| Theorem | bcm1nt 35724 | 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 35725* | 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 35726* | 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 35727 | Lemma for iprodefisum 35728. (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 35728* | 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 35729* | 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 35730* | Lemma for faclim 35733. 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 35731* | Lemma for faclim 35733. 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 35732 | Lemma for faclim 35733. 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 35733* | 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 35734* | 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 35735* | 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 35736 | 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 35737 | 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 35738 | 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 35739* | 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 35740* | 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 35741 | 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 35742 | 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 35743* | 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 35744* | 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 35745* | 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 35746 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
| ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) | ||
| Theorem | cnvco2 35747 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
| ⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴) | ||
| Theorem | eldm3 35748 | Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.) |
| ⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅) | ||
| Theorem | elrn3 35749 | Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.) |
| ⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) | ||
| Theorem | pocnv 35750 | 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 35751 | 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 35752* | 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 35753 | 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 35754 | 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 35755 | 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 35756 | 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 35757 | 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 35758 | 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 35759 | 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 35760 | 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 35761 | 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 35762 | 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 35763 | 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 35764 | 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 35765 | 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 | setinds 35766* | Principle of set induction (or E-induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by Scott Fenton, 10-Mar-2011.) |
| ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ⇒ ⊢ 𝜑 | ||
| Theorem | setinds2f 35767* | E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.) |
| ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) ⇒ ⊢ 𝜑 | ||
| Theorem | setinds2 35768* | E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) |
| ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) ⇒ ⊢ 𝜑 | ||
| Theorem | elpotr 35769* | 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 35770 | Given ax-reg 9545, 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 35771 | Lemma for dfon2 35780. (Contributed by Scott Fenton, 28-Feb-2011.) |
| ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} | ||
| Theorem | dfon2lem2 35772* | Lemma for dfon2 35780. (Contributed by Scott Fenton, 28-Feb-2011.) |
| ⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴 | ||
| Theorem | dfon2lem3 35773* | Lemma for dfon2 35780. 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 35774* | Lemma for dfon2 35780. 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 35775* | Lemma for dfon2 35780. 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 35776* | Lemma for dfon2 35780. 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 35777* | Lemma for dfon2 35780. 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 35778* | Lemma for dfon2 35780. 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 35779* | Lemma for dfon2 35780. 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 35780* | 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 35781 | 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 35782 | 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 35783* | 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 35784* | Generalization of dfrdg2 35783 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 35785 | A version of ax-ext 2701 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 35786 | A version of ax-8 2111 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.) |
| ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦)) | ||
| Theorem | axextdist 35787 | ax-ext 2701 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 35788 | axextb 2704 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 35789* | Version of 19.12vv 2345 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 35790 | There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦 | ||
| Theorem | distel 35791 | Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5397 and elirrv 9549.) (Contributed by Scott Fenton, 15-Dec-2010.) |
| ⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦) | ||
| Theorem | axextndbi 35792 | axextnd 10544 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.) |
| ⊢ ∃𝑧(𝑥 = 𝑦 ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | ||
| Theorem | hbntg 35793 | A more general form of hbnt 2294. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑)) | ||
| Theorem | hbimtg 35794 | A more general and closed form of hbim 2299. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒 → 𝜓) → ∀𝑥(𝜑 → 𝜃))) | ||
| Theorem | hbaltg 35795 | A more general and closed form of hbal 2168. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜓)) | ||
| Theorem | hbng 35796 | A more general form of hbn 2295. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑) | ||
| Theorem | hbimg 35797 | A more general form of hbim 2299. (Contributed by Scott Fenton, 13-Dec-2010.) |
| ⊢ (𝜑 → ∀𝑥𝜓) & ⊢ (𝜒 → ∀𝑥𝜃) ⇒ ⊢ ((𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜃)) | ||
| Syntax | cwsuc 35798 | Declare the syntax for well-founded successor. |
| class wsuc(𝑅, 𝐴, 𝑋) | ||
| Syntax | cwlim 35799 | Declare the syntax for well-founded limit class. |
| class WLim(𝑅, 𝐴) | ||
| Definition | df-wsuc 35800 | Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
| ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | ||
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