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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fpwfvss 43401 | Functions into a powerset always have values which are subsets. This is dependant on our convention when the argument is not part of the domain. (Contributed by RP, 13-Sep-2024.) |
⊢ 𝐹:𝐶⟶𝒫 𝐵 ⇒ ⊢ (𝐹‘𝐴) ⊆ 𝐵 | ||
Theorem | sdomne0 43402 | A class that strictly dominates any set is not empty. (Suggested by SN, 14-Jan-2025.) (Contributed by RP, 14-Jan-2025.) |
⊢ (𝐵 ≺ 𝐴 → 𝐴 ≠ ∅) | ||
Theorem | sdomne0d 43403 | A class that strictly dominates any set is not empty. (Contributed by RP, 3-Sep-2024.) |
⊢ (𝜑 → 𝐵 ≺ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐴 ≠ ∅) | ||
Theorem | safesnsupfiss 43404 | If 𝐵 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.) |
⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o)) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵) | ||
Theorem | safesnsupfiub 43405* | If 𝐵 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.) |
⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o)) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑥𝑅𝑦) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)∀𝑦 ∈ 𝐶 𝑥𝑅𝑦) | ||
Theorem | safesnsupfidom1o 43406 | If 𝐵 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.) |
⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o)) & ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o) | ||
Theorem | safesnsupfilb 43407* | If 𝐵 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 3-Sep-2024.) |
⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o)) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵))∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦) | ||
Theorem | isoeq145d 43408 | Equality deduction for isometries. (Contributed by RP, 14-Jan-2025.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))) | ||
Theorem | resisoeq45d 43409 | Equality deduction for equally restricted isometries. (Contributed by RP, 14-Jan-2025.) |
⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))) | ||
Theorem | negslem1 43410 | An equivalence between identically restricted order-reversing self-isometries. (Contributed by RP, 30-Sep-2024.) |
⊢ (𝐴 = 𝐵 → ((𝐹 ↾ 𝐴) Isom 𝑅, ◡𝑅(𝐴, 𝐴) ↔ (𝐹 ↾ 𝐵) Isom 𝑅, ◡𝑅(𝐵, 𝐵))) | ||
Theorem | nvocnvb 43411* | Equivalence to saying the converse of an involution is the function itself. (Contributed by RP, 13-Oct-2024.) |
⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ↔ (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥)) | ||
Theorem | rp-brsslt 43412* | Binary relation form of a relation, <, which has been extended from relation 𝑅 to subsets of class 𝑆. Usually, we will assume 𝑅 Or 𝑆. Definition in [Alling], p. 2. Generalization of brsslt 27844. (Originally by Scott Fenton, 8-Dec-2021.) (Contributed by RP, 28-Nov-2023.) |
⊢ < = {〈𝑎, 𝑏〉 ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)} ⇒ ⊢ (𝐴 < 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥𝑅𝑦))) | ||
Theorem | nla0002 43413* | Extending a linear order to subsets, the empty set is less than any subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.) |
⊢ < = {〈𝑎, 𝑏〉 ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) ⇒ ⊢ (𝜑 → ∅ < 𝐴) | ||
Theorem | nla0003 43414* | Extending a linear order to subsets, the empty set is greater than any subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.) |
⊢ < = {〈𝑎, 𝑏〉 ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) ⇒ ⊢ (𝜑 → 𝐴 < ∅) | ||
Theorem | nla0001 43415* | Extending a linear order to subsets, the empty set is less than itself. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.) |
⊢ < = {〈𝑎, 𝑏〉 ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)} ⇒ ⊢ (𝜑 → ∅ < ∅) | ||
Theorem | faosnf0.11b 43416* |
𝐵
is called a non-limit ordinal if it is not a limit ordinal.
(Contributed by RP, 27-Sep-2023.)
Alling, Norman L. "Fundamentals of Analysis Over Surreal Numbers Fields." The Rocky Mountain Journal of Mathematics 19, no. 3 (1989): 565-73. http://www.jstor.org/stable/44237243. |
⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) | ||
Theorem | dfno2 43417 | A surreal number, in the functional sign expansion representation, is a function which maps from an ordinal into a set of two possible signs. (Contributed by RP, 12-Jan-2025.) |
⊢ No = {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)} | ||
Theorem | onnog 43418 | Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No ) | ||
Theorem | onnobdayg 43419 | Every ordinal maps to a surreal number of that birthday. (Contributed by RP, 21-Sep-2023.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴) | ||
Theorem | bdaybndex 43420 | Bounds formed from the birthday are surreal numbers. (Contributed by RP, 21-Sep-2023.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No ) | ||
Theorem | bdaybndbday 43421 | Bounds formed from the birthday have the same birthday. (Contributed by RP, 30-Sep-2023.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = ( bday ‘𝐴)) | ||
Theorem | onno 43422 | Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.) |
⊢ (𝐴 ∈ On → (𝐴 × {2o}) ∈ No ) | ||
Theorem | onnoi 43423 | Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐴 × {2o}) ∈ No | ||
Theorem | 0no 43424 | Ordinal zero maps to a surreal number. (Contributed by RP, 21-Sep-2023.) |
⊢ ∅ ∈ No | ||
Theorem | 1no 43425 | Ordinal one maps to a surreal number. (Contributed by RP, 21-Sep-2023.) |
⊢ (1o × {2o}) ∈ No | ||
Theorem | 2no 43426 | Ordinal two maps to a surreal number. (Contributed by RP, 21-Sep-2023.) |
⊢ (2o × {2o}) ∈ No | ||
Theorem | 3no 43427 | Ordinal three maps to a surreal number. (Contributed by RP, 21-Sep-2023.) |
⊢ (3o × {2o}) ∈ No | ||
Theorem | 4no 43428 | Ordinal four maps to a surreal number. (Contributed by RP, 21-Sep-2023.) |
⊢ (4o × {2o}) ∈ No | ||
Theorem | fnimafnex 43429 | The functional image of a function value exists. (Contributed by RP, 31-Oct-2024.) |
⊢ 𝐹 Fn 𝐵 ⇒ ⊢ (𝐹 “ (𝐺‘𝐴)) ∈ V | ||
Theorem | nlimsuc 43430 | A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.) |
⊢ (𝐴 ∈ On → ¬ Lim suc 𝐴) | ||
Theorem | nlim1NEW 43431 | 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.) |
⊢ ¬ Lim 1o | ||
Theorem | nlim2NEW 43432 | 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.) |
⊢ ¬ Lim 2o | ||
Theorem | nlim3 43433 | 3 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.) |
⊢ ¬ Lim 3o | ||
Theorem | nlim4 43434 | 4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.) |
⊢ ¬ Lim 4o | ||
Theorem | oa1un 43435 | Given 𝐴 ∈ On, let 𝐴 +o 1o be defined to be the union of 𝐴 and {𝐴}. Compare with oa1suc 8567. (Contributed by RP, 27-Sep-2023.) |
⊢ (𝐴 ∈ On → (𝐴 +o 1o) = (𝐴 ∪ {𝐴})) | ||
Theorem | oa1cl 43436 | 𝐴 +o 1o is in On. (Contributed by RP, 27-Sep-2023.) |
⊢ (𝐴 ∈ On → (𝐴 +o 1o) ∈ On) | ||
Theorem | 0finon 43437 | 0 is a finite ordinal. See peano1 7910. (Contributed by RP, 27-Sep-2023.) |
⊢ ∅ ∈ (On ∩ Fin) | ||
Theorem | 1finon 43438 | 1 is a finite ordinal. See 1onn 8676. (Contributed by RP, 27-Sep-2023.) |
⊢ 1o ∈ (On ∩ Fin) | ||
Theorem | 2finon 43439 | 2 is a finite ordinal. See 1onn 8676. (Contributed by RP, 27-Sep-2023.) |
⊢ 2o ∈ (On ∩ Fin) | ||
Theorem | 3finon 43440 | 3 is a finite ordinal. See 1onn 8676. (Contributed by RP, 27-Sep-2023.) |
⊢ 3o ∈ (On ∩ Fin) | ||
Theorem | 4finon 43441 | 4 is a finite ordinal. See 1onn 8676. (Contributed by RP, 27-Sep-2023.) |
⊢ 4o ∈ (On ∩ Fin) | ||
Theorem | finona1cl 43442 | The finite ordinals are closed under the add one operation. (Contributed by RP, 27-Sep-2023.) |
⊢ (𝑁 ∈ (On ∩ Fin) → (𝑁 +o 1o) ∈ (On ∩ Fin)) | ||
Theorem | finonex 43443 | The finite ordinals are a set. See also onprc 7796 and fiprc 9083 for proof that On and Fin are proper classes. (Contributed by RP, 27-Sep-2023.) |
⊢ (On ∩ Fin) ∈ V | ||
Theorem | fzunt 43444 | Union of two adjacent finite sets of sequential integers that share a common endpoint. (Suggested by NM, 21-Jul-2005.) (Contributed by RP, 14-Dec-2024.) |
⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁)) | ||
Theorem | fzuntd 43445 | Union of two adjacent finite sets of sequential integers that share a common endpoint. (Contributed by RP, 14-Dec-2024.) |
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁)) | ||
Theorem | fzunt1d 43446 | Union of two overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.) |
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) & ⊢ (𝜑 → 𝑀 ≤ 𝐿) & ⊢ (𝜑 → 𝐿 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁)) | ||
Theorem | fzuntgd 43447 | Union of two adjacent or overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.) |
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) & ⊢ (𝜑 → 𝑀 ≤ (𝐿 + 1)) & ⊢ (𝜑 → 𝐿 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁)) | ||
Theorem | ifpan123g 43448 | Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.) |
⊢ ((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)) ∧ ((¬ 𝜓 ∨ 𝜃) ∧ (𝜓 ∨ 𝜂)))) | ||
Theorem | ifpan23 43449 | Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.) |
⊢ ((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓 ∧ 𝜃), (𝜒 ∧ 𝜏))) | ||
Theorem | ifpdfor2 43450 | Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, 𝜑, 𝜓)) | ||
Theorem | ifporcor 43451 | Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.) |
⊢ (if-(𝜑, 𝜑, 𝜓) ↔ if-(𝜓, 𝜓, 𝜑)) | ||
Theorem | ifpdfan2 43452 | Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, 𝜑)) | ||
Theorem | ifpancor 43453 | Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓)) | ||
Theorem | ifpdfor 43454 | Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, ⊤, 𝜓)) | ||
Theorem | ifpdfan 43455 | Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, ⊥)) | ||
Theorem | ifpbi2 43456 | Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.) |
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃))) | ||
Theorem | ifpbi3 43457 | Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.) |
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓))) | ||
Theorem | ifpim1 43458 | Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 → 𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓)) | ||
Theorem | ifpnot 43459 | Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ (¬ 𝜑 ↔ if-(𝜑, ⊥, ⊤)) | ||
Theorem | ifpid2 43460 | Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ (𝜑 ↔ if-(𝜑, ⊤, ⊥)) | ||
Theorem | ifpim2 43461 | Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑)) | ||
Theorem | ifpbi23 43462 | Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃))) | ||
Theorem | ifpbiidcor 43463 | Restatement of biid 261. (Contributed by RP, 25-Apr-2020.) |
⊢ if-(𝜑, 𝜑, ¬ 𝜑) | ||
Theorem | ifpbicor 43464 | Corollary of commutation of biconditional. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑)) | ||
Theorem | ifpxorcor 43465 | Corollary of commutation of biconditional. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑)) | ||
Theorem | ifpbi1 43466 | Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.) |
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃))) | ||
Theorem | ifpnot23 43467 | Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.) |
⊢ (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒)) | ||
Theorem | ifpnotnotb 43468 | Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ¬ if-(𝜑, 𝜓, 𝜒)) | ||
Theorem | ifpnorcor 43469 | Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, ¬ 𝜑, ¬ 𝜓) ↔ if-(𝜓, ¬ 𝜓, ¬ 𝜑)) | ||
Theorem | ifpnancor 43470 | Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜑) ↔ if-(𝜓, ¬ 𝜑, ¬ 𝜓)) | ||
Theorem | ifpnot23b 43471 | Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒)) | ||
Theorem | ifpbiidcor2 43472 | Restatement of biid 261. (Contributed by RP, 25-Apr-2020.) |
⊢ ¬ if-(𝜑, ¬ 𝜑, 𝜑) | ||
Theorem | ifpnot23c 43473 | Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, 𝜒)) | ||
Theorem | ifpnot23d 43474 | Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒)) | ||
Theorem | ifpdfnan 43475 | Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ⊼ 𝜓) ↔ if-(𝜑, ¬ 𝜓, ⊤)) | ||
Theorem | ifpdfxor 43476 | Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜓)) | ||
Theorem | ifpbi12 43477 | Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜏))) | ||
Theorem | ifpbi13 43478 | Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃))) | ||
Theorem | ifpbi123 43479 | Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂))) | ||
Theorem | ifpidg 43480 | Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜃 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑 ∧ 𝜓) → 𝜃) ∧ ((𝜑 ∧ 𝜃) → 𝜓)) ∧ ((𝜒 → (𝜑 ∨ 𝜃)) ∧ (𝜃 → (𝜑 ∨ 𝜒))))) | ||
Theorem | ifpid3g 43481 | Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ ((𝜑 ∧ 𝜒) → 𝜓))) | ||
Theorem | ifpid2g 43482 | Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜓 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜓 → (𝜑 ∨ 𝜒)) ∧ (𝜒 → (𝜑 ∨ 𝜓)))) | ||
Theorem | ifpid1g 43483 | Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜒 → 𝜑) ∧ (𝜑 → 𝜓))) | ||
Theorem | ifpim23g 43484 | Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (((𝜑 → 𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ (𝜒 → (𝜑 ∨ 𝜓)))) | ||
Theorem | ifpim3 43485 | Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑)) | ||
Theorem | ifpnim1 43486 | Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (¬ (𝜑 → 𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜑)) | ||
Theorem | ifpim4 43487 | Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑)) | ||
Theorem | ifpnim2 43488 | Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (¬ (𝜑 → 𝜓) ↔ if-(𝜓, ¬ 𝜓, 𝜑)) | ||
Theorem | ifpim123g 43489 | Implication of conditional logical operators. The right hand side is basically conjunctive normal form which is useful in proofs. (Contributed by RP, 16-Apr-2020.) |
⊢ ((if-(𝜑, 𝜒, 𝜏) → if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 → 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜏 → 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 → 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜏 → 𝜂))))) | ||
Theorem | ifpim1g 43490 | Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.) |
⊢ ((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ (((𝜓 → 𝜑) ∨ (𝜃 → 𝜒)) ∧ ((𝜑 → 𝜓) ∨ (𝜒 → 𝜃)))) | ||
Theorem | ifp1bi 43491 | Substitute the first element of conditional logical operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)) ↔ ((((𝜑 → 𝜓) ∨ (𝜒 → 𝜃)) ∧ ((𝜑 → 𝜓) ∨ (𝜃 → 𝜒))) ∧ (((𝜓 → 𝜑) ∨ (𝜒 → 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜃 → 𝜒))))) | ||
Theorem | ifpbi1b 43492 | When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, 𝜒, 𝜒) ↔ if-(𝜓, 𝜒, 𝜒)) | ||
Theorem | ifpimimb 43493 | Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpororb 43494 | Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 ∨ 𝜒), (𝜃 ∨ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∨ if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpananb 43495 | Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 ∧ 𝜒), (𝜃 ∧ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∧ if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpnannanb 43496 | Factor conditional logic operator over nand in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 ⊼ 𝜒), (𝜃 ⊼ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ⊼ if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpor123g 43497 | Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.) |
⊢ ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 ∨ 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 ∨ 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜂))))) | ||
Theorem | ifpimim 43498 | Consequnce of implication. (Contributed by RP, 17-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpbibib 43499 | Factor conditional logic operator over biconditional in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 ↔ 𝜒), (𝜃 ↔ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpxorxorb 43500 | Factor conditional logic operator over xor in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 ⊻ 𝜒), (𝜃 ⊻ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ⊻ if-(𝜑, 𝜒, 𝜏))) |
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