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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | or3di 31701 | Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.) |
⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏))) | ||
Theorem | or3dir 31702 | Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∨ 𝜏) ↔ ((𝜑 ∨ 𝜏) ∧ (𝜓 ∨ 𝜏) ∧ (𝜒 ∨ 𝜏))) | ||
Theorem | 3o1cs 31703 | Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | 3o2cs 31704 | Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃) ⇒ ⊢ (𝜓 → 𝜃) | ||
Theorem | 3o3cs 31705 | Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃) ⇒ ⊢ (𝜒 → 𝜃) | ||
Theorem | 13an22anass 31706 | Associative law for four conjunctions with a triple conjunction. (Contributed by Thierry Arnoux, 21-Jan-2025.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃))) | ||
Theorem | sbc2iedf 31707* | Conversion of implicit substitution to explicit class substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑦𝜒 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) | ||
Theorem | rspc2daf 31708* | Double restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑦𝜒 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜓) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | ralcom4f 31709* | Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rexcom4f 31710* | Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | 19.9d2rf 31711 | A deduction version of one direction of 19.9 2199 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | 19.9d2r 31712* | A deduction version of one direction of 19.9 2199 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | r19.29ffa 31713* | A commonly used pattern based on r19.29 3115, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.) |
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒) ⇒ ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) → 𝜒) | ||
Theorem | eqtrb 31714 | A transposition of equality. (Contributed by Thierry Arnoux, 20-Aug-2023.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) | ||
Theorem | eqelbid 31715* | A variable elimination law for equality within a given set 𝐴. See equvel 2456. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | opsbc2ie 31716* | Conversion of implicit substitution to explicit class substitution for ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.) |
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒)) ⇒ ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)) | ||
Theorem | opreu2reuALT 31717* | Correspondence between uniqueness of ordered pairs and double restricted existential uniqueness quantification. Alternate proof of one direction only, use opreu2reurex 6294 instead. (Contributed by Thierry Arnoux, 4-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒)) ⇒ ⊢ ((∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒) → ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑) | ||
Syntax | w2reu 31718 | Syntax for double restricted existential uniqueness quantification. |
wff ∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 | ||
Definition | df-2reu 31719 | Define the double restricted existential uniqueness quantifier. (Contributed by Thierry Arnoux, 4-Jul-2023.) |
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | 2reucom 31720 | Double restricted existential uniqueness commutes. (Contributed by Thierry Arnoux, 4-Jul-2023.) |
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑦 ∈ 𝐵 , 𝑥 ∈ 𝐴𝜑) | ||
Theorem | 2reu2rex1 31721 | Double restricted existential uniqueness implies double restricted existence. (Contributed by Thierry Arnoux, 4-Jul-2023.) |
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) | ||
Theorem | 2reureurex 31722 | Double restricted existential uniqueness implies restricted existential uniqueness with restricted existence. (Contributed by AV, 5-Jul-2023.) |
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) | ||
Theorem | 2reu2reu2 31723* | Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023.) |
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | ||
Theorem | opreu2reu1 31724* | Equivalent definition of the double restricted existential uniqueness quantifier, using uniqueness of ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.) |
⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒 ↔ 𝜑)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒) | ||
Theorem | sq2reunnltb 31725* | There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4𝑘 + 1. Double restricted existential uniqueness variant of 2sqreunnltb 26964. (Contributed by AV, 5-Jul-2023.) |
⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ , 𝑏 ∈ ℕ(𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) | ||
Theorem | addsqnot2reu 31726* | For each complex number 𝐶, there does not uniquely exist two complex numbers 𝑎 and 𝑏, with 𝑏 squared and added to 𝑎 resulting in the given complex number 𝐶. Double restricted existential uniqueness variant of addsqn2reurex2 26948. (Contributed by AV, 5-Jul-2023.) |
⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑎 ∈ ℂ , 𝑏 ∈ ℂ(𝑎 + (𝑏↑2)) = 𝐶) | ||
Theorem | sbceqbidf 31727 | Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) | ||
Theorem | sbcies 31728* | A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
⊢ 𝐴 = (𝐸‘𝑊) & ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑)) | ||
Theorem | mo5f 31729* | Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.) |
⊢ Ⅎ𝑖𝜑 & ⊢ Ⅎ𝑗𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑖∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗)) | ||
Theorem | nmo 31730* | Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (¬ ∃*𝑥𝜑 ↔ ∀𝑦∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦)) | ||
Theorem | reuxfrdf 31731* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 3745 (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) (Revised by Thierry Arnoux, 30-Mar-2018.) |
⊢ Ⅎ𝑦𝐵 & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓)) | ||
Theorem | rexunirn 31732* | Restricted existential quantification over the union of the range of a function. Cf. rexrn 7089 and eluni2 4913. (Contributed by Thierry Arnoux, 19-Sep-2017.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ ∪ ran 𝐹𝜑) | ||
Theorem | rmoxfrd 31733* | Transfer "at most one" restricted quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | rmoun 31734 | "At most one" restricted existential quantifier for a union implies the same quantifier on both sets. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 → (∃*𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | rmounid 31735* | A case where an "at most one" restricted existential quantifier for a union is equivalent to such a quantifier for one of the sets. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝜓) ⇒ ⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜓)) | ||
Theorem | riotaeqbidva 31736* | Equivalent wff's yield equal restricted definition binders (deduction form). (raleqbidva 3328 analog.) (Contributed by Thierry Arnoux, 29-Jan-2025.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒)) | ||
Theorem | dmrab 31737* | Domain of a restricted class abstraction over a cartesian product. (Contributed by Thierry Arnoux, 3-Jul-2023.) |
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓)) ⇒ ⊢ dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓} | ||
Theorem | difrab2 31738 | Difference of two restricted class abstractions. Compare with difrab 4309. (Contributed by Thierry Arnoux, 3-Jan-2022.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐵 ∣ 𝜑}) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} | ||
Theorem | rabexgfGS 31739 | Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | rabsnel 31740* | Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ 𝐴) | ||
Theorem | eqrrabd 31741* | Deduce equality with a restricted abstraction. (Contributed by Thierry Arnoux, 11-Apr-2024.) |
⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓)) ⇒ ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}) | ||
Theorem | foresf1o 31742* | From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵) | ||
Theorem | rabfodom 31743* | Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵) ⇒ ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓}) | ||
Theorem | abrexdomjm 31744* | An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑) ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴) | ||
Theorem | abrexdom2jm 31745* | An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ≼ 𝐴) | ||
Theorem | abrexexd 31746* | Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝐴 ∈ V) ⇒ ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) | ||
Theorem | elabreximd 31747* | Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) | ||
Theorem | elabreximdv 31748* | Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) |
⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) | ||
Theorem | abrexss 31749* | A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.) |
⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶) | ||
Theorem | elunsn 31750 | Elementhood to a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶))) | ||
Theorem | nelun 31751 | Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.) |
⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶))) | ||
Theorem | snsssng 31752 | If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.) (Revised by Thierry Arnoux, 11-Apr-2024.) |
⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵) | ||
Theorem | inin 31753 | Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.) |
⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵) | ||
Theorem | inindif 31754 | See inundif 4479. (Contributed by Thierry Arnoux, 13-Sep-2017.) |
⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅ | ||
Theorem | difininv 31755 | Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021.) |
⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = (𝐶 ∩ 𝐵)) | ||
Theorem | difeq 31756 | Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
⊢ ((𝐴 ∖ 𝐵) = 𝐶 ↔ ((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵))) | ||
Theorem | eqdif 31757 | If both set differences of two sets are empty, those sets are equal. (Contributed by Thierry Arnoux, 16-Nov-2023.) |
⊢ (((𝐴 ∖ 𝐵) = ∅ ∧ (𝐵 ∖ 𝐴) = ∅) → 𝐴 = 𝐵) | ||
Theorem | indifbi 31758 | Two ways to express equality relative to a class 𝐴. (Contributed by Thierry Arnoux, 23-Jun-2024.) |
⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝐶)) | ||
Theorem | diffib 31759 | Case where diffi 9179 is a biconditional. (Contributed by Thierry Arnoux, 27-Jun-2024.) |
⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin ↔ (𝐴 ∖ 𝐵) ∈ Fin)) | ||
Theorem | difxp1ss 31760 | Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.) |
⊢ ((𝐴 ∖ 𝐶) × 𝐵) ⊆ (𝐴 × 𝐵) | ||
Theorem | difxp2ss 31761 | Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.) |
⊢ (𝐴 × (𝐵 ∖ 𝐶)) ⊆ (𝐴 × 𝐵) | ||
Theorem | indifundif 31762 | A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.) |
⊢ (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐵 ∩ 𝐶)) | ||
Theorem | elpwincl1 31763 | Closure of intersection with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶) | ||
Theorem | elpwdifcl 31764 | Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐶) | ||
Theorem | elpwiuncl 31765* | Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝒫 𝐶) ⇒ ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶) | ||
Theorem | eqsnd 31766* | Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = {𝐵}) | ||
Theorem | elpreq 31767 | Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
⊢ (𝜑 → 𝑋 ∈ {𝐴, 𝐵}) & ⊢ (𝜑 → 𝑌 ∈ {𝐴, 𝐵}) & ⊢ (𝜑 → (𝑋 = 𝐴 ↔ 𝑌 = 𝐴)) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
Theorem | nelpr 31768 | A set 𝐴 not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶))) | ||
Theorem | inpr0 31769 | Rewrite an empty intersection with a pair. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
⊢ ((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴)) | ||
Theorem | neldifpr1 31770 | The first element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
⊢ ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) | ||
Theorem | neldifpr2 31771 | The second element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
⊢ ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) | ||
Theorem | unidifsnel 31772 | The other element of a pair is an element of the pair. (Contributed by Thierry Arnoux, 26-Aug-2017.) |
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃) | ||
Theorem | unidifsnne 31773 | The other element of a pair is not the known element. (Contributed by Thierry Arnoux, 26-Aug-2017.) |
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋) | ||
Theorem | ifeqeqx 31774* | An equality theorem tailored for ballotlemsf1o 33512. (Contributed by Thierry Arnoux, 14-Apr-2017.) |
⊢ (𝑥 = 𝑋 → 𝐴 = 𝐶) & ⊢ (𝑥 = 𝑌 → 𝐵 = 𝑎) & ⊢ (𝑥 = 𝑋 → (𝜒 ↔ 𝜃)) & ⊢ (𝑥 = 𝑌 → (𝜒 ↔ 𝜓)) & ⊢ (𝜑 → 𝑎 = 𝐶) & ⊢ ((𝜑 ∧ 𝜓) → 𝜃) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑊) ⇒ ⊢ ((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) → 𝑎 = if(𝜒, 𝐴, 𝐵)) | ||
Theorem | elimifd 31775 | Elimination of a conditional operator contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
⊢ (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒 ↔ 𝜃))) & ⊢ (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒 ↔ 𝜏))) ⇒ ⊢ (𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜃) ∨ (¬ 𝜓 ∧ 𝜏)))) | ||
Theorem | elim2if 31776 | Elimination of two conditional operators contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒 ↔ 𝜃)) & ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒 ↔ 𝜏)) & ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒 ↔ 𝜂)) ⇒ ⊢ (𝜒 ↔ ((𝜑 ∧ 𝜃) ∨ (¬ 𝜑 ∧ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂))))) | ||
Theorem | elim2ifim 31777 | Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒 ↔ 𝜃)) & ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒 ↔ 𝜏)) & ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒 ↔ 𝜂)) & ⊢ (𝜑 → 𝜃) & ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜏) & ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂) ⇒ ⊢ 𝜒 | ||
Theorem | ifeq3da 31778 | Given an expression 𝐶 containing if(𝜓, 𝐸, 𝐹), substitute (hypotheses .1 and .2) and evaluate (hypotheses .3 and .4) it for both cases at the same time. (Contributed by Thierry Arnoux, 13-Dec-2021.) |
⊢ (if(𝜓, 𝐸, 𝐹) = 𝐸 → 𝐶 = 𝐺) & ⊢ (if(𝜓, 𝐸, 𝐹) = 𝐹 → 𝐶 = 𝐻) & ⊢ (𝜑 → 𝐺 = 𝐴) & ⊢ (𝜑 → 𝐻 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶) | ||
Theorem | ifnetrue 31779 | Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑) | ||
Theorem | ifnefals 31780 | Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑) | ||
Theorem | ifnebib 31781 | The converse of ifbi 4551 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓))) | ||
Theorem | uniinn0 31782* | Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.) |
⊢ ((∪ 𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅) | ||
Theorem | uniin1 31783* | Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.) |
⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝐴 ∩ 𝐵) | ||
Theorem | uniin2 31784* | Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.) |
⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝐵) | ||
Theorem | difuncomp 31785 | Express a class difference using unions and class complements. (Contributed by Thierry Arnoux, 21-Jun-2020.) |
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∖ 𝐵) = (𝐶 ∖ ((𝐶 ∖ 𝐴) ∪ 𝐵))) | ||
Theorem | elpwunicl 31786 | Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.) (Proof shortened by BJ, 6-Apr-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | cbviunf 31787* | Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | iuneq12daf 31788 | Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) | ||
Theorem | iunin1f 31789 | Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5062 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.) |
⊢ Ⅎ𝑥𝐶 ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ 𝐶) | ||
Theorem | ssiun3 31790* | Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.) |
⊢ (∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | ssiun2sf 31791 | Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ Ⅎ𝑥𝐷 & ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iuninc 31792* | The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.) |
⊢ (𝜑 → 𝐹 Fn ℕ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))) ⇒ ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖)) | ||
Theorem | iundifdifd 31793* | The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.) |
⊢ (𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)))) | ||
Theorem | iundifdif 31794* | The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 31793. (Contributed by Thierry Arnoux, 4-Sep-2016.) |
⊢ 𝑂 ∈ V & ⊢ 𝐴 ⊆ 𝒫 𝑂 ⇒ ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))) | ||
Theorem | iunrdx 31795* | Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.) |
⊢ (𝜑 → 𝐹:𝐴–onto→𝐶) & ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷) | ||
Theorem | iunpreima 31796* | Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵)) | ||
Theorem | iunrnmptss 31797* | A subset relation for an indexed union over the range of function expressed as a mapping. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∪ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐷) | ||
Theorem | iunxunsn 31798* | Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) ⇒ ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)) | ||
Theorem | iunxunpr 31799* | Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) & ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷))) | ||
Theorem | iinabrex 31800* | Rewriting an indexed intersection into an intersection of its image set. (Contributed by Thierry Arnoux, 15-Jun-2024.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
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