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Date | Label | Description |
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Theorem | ||
1-Oct-2024 | infex2g 6978 | Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.) |
⊢ (𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V) | ||
30-Sep-2024 | unbendc 12185 | An unbounded decidable set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Jim Kingdon, 30-Sep-2024.) |
⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ ℕ) | ||
30-Sep-2024 | prmdc 12006 | Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.) |
⊢ (𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ) | ||
30-Sep-2024 | dcfi 6925 | Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.) |
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∀𝑥 ∈ 𝐴 𝜑) | ||
29-Sep-2024 | ssnnct 12176 | A decidable subset of ℕ is countable. (Contributed by Jim Kingdon, 29-Sep-2024.) |
⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1_{o})) | ||
29-Sep-2024 | ssnnctlemct 12175 | Lemma for ssnnct 12176. The result. (Contributed by Jim Kingdon, 29-Sep-2024.) |
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 1) ⇒ ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1_{o})) | ||
28-Sep-2024 | nninfdcex 11838 | A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.) |
⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) & ⊢ (𝜑 → ∃𝑦 𝑦 ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | ||
27-Sep-2024 | infregelbex 9509 | Any lower bound of a set of real numbers with an infimum is less than or equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.) |
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)) | ||
26-Sep-2024 | nninfdclemp1 12181 | Lemma for nninfdc 12184. Each element of the sequence 𝐹 is greater than the previous element. (Contributed by Jim Kingdon, 26-Sep-2024.) |
⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) & ⊢ (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽)) & ⊢ 𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ_{≥}‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽)) & ⊢ (𝜑 → 𝑈 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐹‘𝑈) < (𝐹‘(𝑈 + 1))) | ||
26-Sep-2024 | nnminle 12178 | The infimum of a decidable subset of the natural numbers is less than an element of the set. The infimum is also a minimum as shown at nnmindc 12177. (Contributed by Jim Kingdon, 26-Sep-2024.) |
⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵) | ||
25-Sep-2024 | nninfdclemcl 12179 | Lemma for nninfdc 12184. (Contributed by Jim Kingdon, 25-Sep-2024.) |
⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑃(𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ_{≥}‘(𝑦 + 1))), ℝ, < ))𝑄) ∈ 𝐴) | ||
24-Sep-2024 | nninfdclemlt 12182 | Lemma for nninfdc 12184. The function from nninfdclemf 12180 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.) |
⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) & ⊢ (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽)) & ⊢ 𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ_{≥}‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽)) & ⊢ (𝜑 → 𝑈 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ ℕ) & ⊢ (𝜑 → 𝑈 < 𝑉) ⇒ ⊢ (𝜑 → (𝐹‘𝑈) < (𝐹‘𝑉)) | ||
23-Sep-2024 | nninfdc 12184 | An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.) |
⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → ω ≼ 𝐴) | ||
23-Sep-2024 | nninfdclemf1 12183 | Lemma for nninfdc 12184. The function from nninfdclemf 12180 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.) |
⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) & ⊢ (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽)) & ⊢ 𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ_{≥}‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽)) ⇒ ⊢ (𝜑 → 𝐹:ℕ–1-1→𝐴) | ||
23-Sep-2024 | nninfdclemf 12180 | Lemma for nninfdc 12184. A function from the natural numbers into 𝐴. (Contributed by Jim Kingdon, 23-Sep-2024.) |
⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) & ⊢ (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽)) & ⊢ 𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ_{≥}‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽)) ⇒ ⊢ (𝜑 → 𝐹:ℕ⟶𝐴) | ||
23-Sep-2024 | nnmindc 12177 | An inhabited decidable subset of the natural numbers has a minimum. (Contributed by Jim Kingdon, 23-Sep-2024.) |
⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐴) → inf(𝐴, ℝ, < ) ∈ 𝐴) | ||
19-Sep-2024 | ssomct 12174 | A decidable subset of ω is countable. (Contributed by Jim Kingdon, 19-Sep-2024.) |
⊢ ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1_{o})) | ||
14-Sep-2024 | nnpredlt 4583 | The predecessor (see nnpredcl 4582) of a nonzero natural number is less than (see df-iord 4326) that number. (Contributed by Jim Kingdon, 14-Sep-2024.) |
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴) | ||
13-Sep-2024 | nninfisollemeq 7075 | Lemma for nninfisol 7076. The case where 𝑁 is a successor and 𝑁 and 𝑋 are equal. (Contributed by Jim Kingdon, 13-Sep-2024.) |
⊢ (𝜑 → 𝑋 ∈ ℕ_{∞}) & ⊢ (𝜑 → (𝑋‘𝑁) = ∅) & ⊢ (𝜑 → 𝑁 ∈ ω) & ⊢ (𝜑 → 𝑁 ≠ ∅) & ⊢ (𝜑 → (𝑋‘∪ 𝑁) = 1_{o}) ⇒ ⊢ (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1_{o}, ∅)) = 𝑋) | ||
13-Sep-2024 | nninfisollemne 7074 | Lemma for nninfisol 7076. A case where 𝑁 is a successor and 𝑁 and 𝑋 are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.) |
⊢ (𝜑 → 𝑋 ∈ ℕ_{∞}) & ⊢ (𝜑 → (𝑋‘𝑁) = ∅) & ⊢ (𝜑 → 𝑁 ∈ ω) & ⊢ (𝜑 → 𝑁 ≠ ∅) & ⊢ (𝜑 → (𝑋‘∪ 𝑁) = ∅) ⇒ ⊢ (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1_{o}, ∅)) = 𝑋) | ||
13-Sep-2024 | nninfisollem0 7073 | Lemma for nninfisol 7076. The case where 𝑁 is zero. (Contributed by Jim Kingdon, 13-Sep-2024.) |
⊢ (𝜑 → 𝑋 ∈ ℕ_{∞}) & ⊢ (𝜑 → (𝑋‘𝑁) = ∅) & ⊢ (𝜑 → 𝑁 ∈ ω) & ⊢ (𝜑 → 𝑁 = ∅) ⇒ ⊢ (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1_{o}, ∅)) = 𝑋) | ||
12-Sep-2024 | nninfisol 7076 | Finite elements of ℕ_{∞} are isolated. That is, given a natural number and any element of ℕ_{∞}, it is decidable whether the natural number (when converted to an element of ℕ_{∞}) is equal to the given element of ℕ_{∞}. Stated in an online post by Martin Escardo. One way to understand this theorem is that you do not need to look at an unbounded number of elements of the sequence 𝑋 to decide whether it is equal to 𝑁 (in fact, you only need to look at two elements and 𝑁 tells you where to look). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.) |
⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ_{∞}) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1_{o}, ∅)) = 𝑋) | ||
7-Sep-2024 | eulerthlemfi 12102 | Lemma for eulerth 12107. The set 𝑆 is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.) |
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) & ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⇒ ⊢ (𝜑 → 𝑆 ∈ Fin) | ||
7-Sep-2024 | modqexp 10544 | Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℕ_{0}) & ⊢ (𝜑 → 𝐷 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐷) & ⊢ (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ⇒ ⊢ (𝜑 → ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷)) | ||
5-Sep-2024 | eulerthlemh 12105 | Lemma for eulerth 12107. A permutation of (1...(ϕ‘𝑁)). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-Sep-2024.) |
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) & ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} & ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆) & ⊢ 𝐻 = (^{◡}𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))) ⇒ ⊢ (𝜑 → 𝐻:(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁))) | ||
2-Sep-2024 | eulerthlemth 12106 | Lemma for eulerth 12107. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.) |
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) & ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} & ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆) ⇒ ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁)) | ||
2-Sep-2024 | eulerthlema 12104 | Lemma for eulerth 12107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.) |
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) & ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} & ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆) ⇒ ⊢ (𝜑 → (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) mod 𝑁) = (∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) mod 𝑁)) | ||
2-Sep-2024 | eulerthlemrprm 12103 | Lemma for eulerth 12107. 𝑁 and ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.) |
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) & ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} & ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆) ⇒ ⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1) | ||
30-Aug-2024 | fprodap0f 11533 | A finite product of terms apart from zero is apart from zero. A version of fprodap0 11518 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 # 0) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 # 0) | ||
28-Aug-2024 | fprodrec 11526 | The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 # 0) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵)) | ||
26-Aug-2024 | exmidontri2or 7178 | Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
⊢ (EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
26-Aug-2024 | exmidontri 7174 | Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
⊢ (EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
26-Aug-2024 | ontri2orexmidim 4531 | Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4530. (Contributed by Jim Kingdon, 26-Aug-2024.) |
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝜑) | ||
26-Aug-2024 | ontriexmidim 4481 | Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4480. (Contributed by Jim Kingdon, 26-Aug-2024.) |
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → DECID 𝜑) | ||
25-Aug-2024 | onntri2or 7181 | Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.) |
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
25-Aug-2024 | onntri3or 7180 | Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.) |
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
25-Aug-2024 | csbcow 3042 | Composition law for chained substitutions into a class. Version of csbco 3041 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 25-Aug-2024.) |
⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 | ||
25-Aug-2024 | cbvreuvw 2686 | Version of cbvreuv 2682 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | ||
25-Aug-2024 | cbvrexvw 2685 | Version of cbvrexv 2681 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) | ||
25-Aug-2024 | cbvralvw 2684 | Version of cbvralv 2680 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | ||
25-Aug-2024 | cbvabw 2280 | Version of cbvab 2281 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | ||
25-Aug-2024 | nfsbv 1927 | If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is distinct from 𝑥 and 𝑦. Version of nfsb 1926 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | ||
25-Aug-2024 | cbvexvw 1900 | Change bound variable. See cbvexv 1898 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1428. (Revised by Gino Giotto, 25-Aug-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
25-Aug-2024 | cbvalvw 1899 | Change bound variable. See cbvalv 1897 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1428. (Revised by Gino Giotto, 25-Aug-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
25-Aug-2024 | nfal 1556 | If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-4 1490. (Revised by Gino Giotto, 25-Aug-2024.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∀𝑦𝜑 | ||
24-Aug-2024 | gcdcomd 11857 | The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) | ||
17-Aug-2024 | fprodcl2lem 11502 | Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ≠ ∅) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) | ||
16-Aug-2024 | if0ab 13380 |
Expression of a conditional class as a class abstraction when the False
alternative is the empty class: in that case, the conditional class is
the extension, in the True alternative, of the condition.
Remark: a consequence which could be formalized is the inclusion ⊢ if(𝜑, 𝐴, ∅) ⊆ 𝐴 and therefore, using elpwg 3551, ⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 13381 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.) |
⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑} | ||
16-Aug-2024 | fprodunsn 11501 | Multiply in an additional term in a finite product. See also fprodsplitsn 11530 which is the same but with a Ⅎ𝑘𝜑 hypothesis in place of the distinct variable condition between 𝜑 and 𝑘. (Contributed by Jim Kingdon, 16-Aug-2024.) |
⊢ Ⅎ𝑘𝐷 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷)) | ||
15-Aug-2024 | bj-charfundcALT 13384 | Alternate proof of bj-charfundc 13383. It was expected to be much shorter since it uses bj-charfun 13382 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1_{o}, ∅))) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹:𝑋⟶2_{o} ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1_{o} ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅))) | ||
15-Aug-2024 | bj-charfun 13382 | Properties of the characteristic function on the class 𝑋 of the class 𝐴. (Contributed by BJ, 15-Aug-2024.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1_{o}, ∅))) ⇒ ⊢ (𝜑 → ((𝐹:𝑋⟶𝒫 1_{o} ∧ (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2_{o}) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1_{o} ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅))) | ||
15-Aug-2024 | fmelpw1o 13381 |
With a formula 𝜑 one can associate an element of
𝒫 1_{o}, which
can therefore be thought of as the set of "truth values" (but
recall that
there are no other genuine truth values than ⊤ and ⊥, by
nndc 837, which translate to 1_{o} and ∅
respectively by iftrue 3510
and iffalse 3513, giving pwtrufal 13569).
As proved in if0ab 13380, the associated element of 𝒫 1_{o} is the extension, in 𝒫 1_{o}, of the formula 𝜑. (Contributed by BJ, 15-Aug-2024.) |
⊢ if(𝜑, 1_{o}, ∅) ∈ 𝒫 1_{o} | ||
15-Aug-2024 | cnstab 8520 | Equality of complex numbers is stable. Stability here means ¬ ¬ 𝐴 = 𝐵 → 𝐴 = 𝐵 as defined at df-stab 817. This theorem for real numbers is Proposition 5.2 of [BauerHanson], p. 27. (Contributed by Jim Kingdon, 1-Aug-2023.) (Proof shortened by BJ, 15-Aug-2024.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → STAB 𝐴 = 𝐵) | ||
15-Aug-2024 | subap0d 8519 | Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.) (Proof shortened by BJ, 15-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) # 0) | ||
15-Aug-2024 | ifexd 4444 | Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) ∈ V) | ||
15-Aug-2024 | ifelpwun 4443 | Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵) | ||
15-Aug-2024 | ifelpwund 4442 | Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)) | ||
15-Aug-2024 | ifelpwung 4441 | Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)) | ||
15-Aug-2024 | ifidss 3520 | A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.) |
⊢ if(𝜑, 𝐴, 𝐴) ⊆ 𝐴 | ||
15-Aug-2024 | ifssun 3519 | A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.) |
⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵) | ||
12-Aug-2024 | exmidontriimlem2 7157 | Lemma for exmidontriim 7160. (Contributed by Jim Kingdon, 12-Aug-2024.) |
⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → EXMID) & ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ ∀𝑦 ∈ 𝐵 𝑦 ∈ 𝐴)) | ||
12-Aug-2024 | exmidontriimlem1 7156 | Lemma for exmidontriim 7160. A variation of r19.30dc 2604. (Contributed by Jim Kingdon, 12-Aug-2024.) |
⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ∧ EXMID) → (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜒)) | ||
11-Aug-2024 | nndc 837 |
Double negation of decidability of a formula. Intuitionistic logic
refutes the negation of decidability (but does not prove decidability) of
any formula.
This should not trick the reader into thinking that ¬ ¬ EXMID is provable in intuitionistic logic. Indeed, if we could quantify over formula metavariables, then generalizing nnexmid 836 over 𝜑 would give "⊢ ∀𝜑¬ ¬ DECID 𝜑", but EXMID is "∀𝜑DECID 𝜑", so proving ¬ ¬ EXMID would amount to proving "¬ ¬ ∀𝜑DECID 𝜑", which is not implied by the above theorem. Indeed, the converse of nnal 1629 does not hold. Since our system does not allow quantification over formula metavariables, we can reproduce this argument by representing formulas as subsets of 𝒫 1_{o}, like we do in our definition of EXMID (df-exmid 4156): then, we can prove ∀𝑥 ∈ 𝒫 1_{o}¬ ¬ DECID 𝑥 = 1_{o} but we cannot prove ¬ ¬ ∀𝑥 ∈ 𝒫 1_{o}DECID 𝑥 = 1_{o} because the converse of nnral 2447 does not hold. Actually, ¬ ¬ EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying ¬ EXMID and noncontradiction holds (pm3.24 683). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of ¬ ¬ EXMID. (Revised by BJ, 11-Aug-2024.) |
⊢ ¬ ¬ DECID 𝜑 | ||
10-Aug-2024 | exmidontriim 7160 | Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.) |
⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
10-Aug-2024 | exmidontriimlem4 7159 | Lemma for exmidontriim 7160. The induction step for the induction on 𝐴. (Contributed by Jim Kingdon, 10-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → EXMID) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | ||
10-Aug-2024 | exmidontriimlem3 7158 | Lemma for exmidontriim 7160. What we get to do based on induction on both 𝐴 and 𝐵. (Contributed by Jim Kingdon, 10-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → EXMID) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)) & ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | ||
10-Aug-2024 | nnnninf2 7070 | Canonical embedding of suc ω into ℕ_{∞}. (Contributed by BJ, 10-Aug-2024.) |
⊢ (𝑁 ∈ suc ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1_{o}, ∅)) ∈ ℕ_{∞}) | ||
10-Aug-2024 | infnninf 7067 | The point at infinity in ℕ_{∞} is the constant sequence equal to 1_{o}. Note that with our encoding of functions, that constant function can also be expressed as (ω × {1_{o}}), as fconstmpt 4633 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.) |
⊢ (𝑖 ∈ ω ↦ 1_{o}) ∈ ℕ_{∞} | ||
9-Aug-2024 | ss1o0el1o 6857 | Reformulation of ss1o0el1 4158 using 1_{o} instead of {∅}. (Contributed by BJ, 9-Aug-2024.) |
⊢ (𝐴 ⊆ 1_{o} → (∅ ∈ 𝐴 ↔ 𝐴 = 1_{o})) | ||
9-Aug-2024 | pw1dc0el 6856 | Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.) |
⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1_{o}DECID ∅ ∈ 𝑥) | ||
9-Aug-2024 | ss1o0el1 4158 | A subclass of {∅} contains the empty set if and only if it equals {∅}. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.) |
⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 ↔ 𝐴 = {∅})) | ||
8-Aug-2024 | pw1dc1 6858 | If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.) |
⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1_{o}DECID 𝑥 = 1_{o}) | ||
7-Aug-2024 | pw1fin 6855 | Excluded middle is equivalent to the power set of 1_{o} being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.) |
⊢ (EXMID ↔ 𝒫 1_{o} ∈ Fin) | ||
7-Aug-2024 | elomssom 4564 | A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4565. (Revised by BJ, 7-Aug-2024.) |
⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) | ||
6-Aug-2024 | bj-charfunbi 13386 |
In an ambient set 𝑋, if membership in 𝐴 is
stable, then it is
decidable if and only if 𝐴 has a characteristic function.
This characterization can be applied to singletons when the set 𝑋 has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 STAB 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∃𝑓 ∈ (2_{o} ↑_{𝑚} 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1_{o} ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))) | ||
6-Aug-2024 | bj-charfunr 13385 |
If a class 𝐴 has a "weak"
characteristic function on a class 𝑋,
then negated membership in 𝐴 is decidable (in other words,
membership in 𝐴 is testable) in 𝑋.
The hypothesis imposes that 𝑋 be a set. As usual, it could be formulated as ⊢ (𝜑 → (𝐹:𝑋⟶ω ∧ ...)) to deal with general classes, but that extra generality would not make the theorem much more useful. The theorem would still hold if the codomain of 𝑓 were any class with testable equality to the point where (𝑋 ∖ 𝐴) is sent. (Contributed by BJ, 6-Aug-2024.) |
⊢ (𝜑 → ∃𝑓 ∈ (ω ↑_{𝑚} 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴) | ||
6-Aug-2024 | bj-charfundc 13383 | Properties of the characteristic function on the class 𝑋 of the class 𝐴, provided membership in 𝐴 is decidable in 𝑋. (Contributed by BJ, 6-Aug-2024.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1_{o}, ∅))) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹:𝑋⟶2_{o} ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1_{o} ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅))) | ||
6-Aug-2024 | prodssdc 11486 | Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.) (Revised by Jim Kingdon, 6-Aug-2024.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ (𝜑 → ∃𝑛 ∈ (ℤ_{≥}‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ (ℤ_{≥}‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1))) ⇝ 𝑦)) & ⊢ (𝜑 → ∀𝑗 ∈ (ℤ_{≥}‘𝑀)DECID 𝑗 ∈ 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 1) & ⊢ (𝜑 → 𝐵 ⊆ (ℤ_{≥}‘𝑀)) & ⊢ (𝜑 → ∀𝑗 ∈ (ℤ_{≥}‘𝑀)DECID 𝑗 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) | ||
5-Aug-2024 | fnmptd 13379 | The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 Fn 𝐴) | ||
5-Aug-2024 | funmptd 13378 |
The maps-to notation defines a function (deduction form).
Note: one should similarly prove a deduction form of funopab4 5207, then prove funmptd 13378 from it, and then prove funmpt 5208 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
5-Aug-2024 | 2ssom 13377 | The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.) |
⊢ 2_{o} ⊆ ω | ||
5-Aug-2024 | bj-dcfal 13327 | The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.) |
⊢ DECID ⊥ | ||
5-Aug-2024 | bj-dctru 13325 | The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.) |
⊢ DECID ⊤ | ||
5-Aug-2024 | bj-stfal 13317 | The false truth value is stable. (Contributed by BJ, 5-Aug-2024.) |
⊢ STAB ⊥ | ||
5-Aug-2024 | bj-sttru 13315 | The true truth value is stable. (Contributed by BJ, 5-Aug-2024.) |
⊢ STAB ⊤ | ||
5-Aug-2024 | prod1dc 11483 | Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.) (Revised by Jim Kingdon, 5-Aug-2024.) |
⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ_{≥}‘𝑀) ∧ ∀𝑗 ∈ (ℤ_{≥}‘𝑀)DECID 𝑗 ∈ 𝐴) ∨ 𝐴 ∈ Fin) → ∏𝑘 ∈ 𝐴 1 = 1) | ||
2-Aug-2024 | onntri52 7179 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
2-Aug-2024 | onntri24 7177 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
2-Aug-2024 | onntri45 7176 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ EXMID) | ||
2-Aug-2024 | onntri51 7175 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
2-Aug-2024 | onntri13 7173 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
2-Aug-2024 | onntri35 7172 |
Double negated ordinal trichotomy.
There are five equivalent statements: (1) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (2) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), (3) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (4) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), and (5) ¬ ¬ EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7173), (3) implies (5) (onntri35 7172), (5) implies (1) (onntri51 7175), (2) implies (4) (onntri24 7177), (4) implies (5) (onntri45 7176), and (5) implies (2) (onntri52 7179). Another way of stating this is that EXMID is equivalent to trichotomy, either the 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 or the 𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥 form, as shown in exmidontri 7174 and exmidontri2or 7178, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7180 and (4) by onntri2or 7181. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID) | ||
1-Aug-2024 | nnral 2447 | The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1629. (Contributed by Jim Kingdon, 1-Aug-2024.) |
⊢ (¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑) | ||
31-Jul-2024 | 3nsssucpw1 7171 | Negated excluded middle implies that 3_{o} is not a subset of the successor of the power set of 1_{o}. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.) |
⊢ (¬ EXMID → ¬ 3_{o} ⊆ suc 𝒫 1_{o}) | ||
31-Jul-2024 | sucpw1nss3 7170 | Negated excluded middle implies that the successor of the power set of 1_{o} is not a subset of 3_{o}. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.) |
⊢ (¬ EXMID → ¬ suc 𝒫 1_{o} ⊆ 3_{o}) | ||
30-Jul-2024 | 3nelsucpw1 7169 | Three is not an element of the successor of the power set of 1_{o}. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ ¬ 3_{o} ∈ suc 𝒫 1_{o} | ||
30-Jul-2024 | sucpw1nel3 7168 | The successor of the power set of 1_{o} is not an element of 3_{o}. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ ¬ suc 𝒫 1_{o} ∈ 3_{o} | ||
30-Jul-2024 | sucpw1ne3 7167 | Negated excluded middle implies that the successor of the power set of 1_{o} is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ (¬ EXMID → suc 𝒫 1_{o} ≠ 3_{o}) | ||
30-Jul-2024 | pw1nel3 7166 | Negated excluded middle implies that the power set of 1_{o} is not an element of 3_{o}. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ (¬ EXMID → ¬ 𝒫 1_{o} ∈ 3_{o}) | ||
30-Jul-2024 | pw1ne3 7165 | The power set of 1_{o} is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ 𝒫 1_{o} ≠ 3_{o} |
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