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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ssun3 4001 | Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | ||
Theorem | ssun4 4002 | Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵)) | ||
Theorem | elun1 4003 | Membership law for union of classes. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶)) | ||
Theorem | elun2 4004 | Membership law for union of classes. (Contributed by NM, 30-Aug-1993.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵)) | ||
Theorem | unss1 4005 | Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | ||
Theorem | ssequn1 4006 | A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) | ||
Theorem | unss2 4007 | Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∪ 𝐴) ⊆ (𝐶 ∪ 𝐵)) | ||
Theorem | unss12 4008 | Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) | ||
Theorem | ssequn2 4009 | A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) | ||
Theorem | unss 4010 | The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | ||
Theorem | unssi 4011 | An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝐴 ⊆ 𝐶 & ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶 | ||
Theorem | unssd 4012 | A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) | ||
Theorem | unssad 4013 | If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 4010. Partial converse of unssd 4012. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | unssbd 4014 | If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 4010. Partial converse of unssd 4012. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | ssun 4015 | A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.) |
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | ||
Theorem | rexun 4016 | Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.) |
⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | ralunb 4017 | Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | ralun 4018 | Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑) | ||
Theorem | elin 4019 | Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | ||
Theorem | elini 4020 | Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) | ||
Theorem | elind 4021 | Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) | ||
Theorem | elinel1 4022 | Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵) | ||
Theorem | elinel2 4023 | Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐶) | ||
Theorem | elin2 4024 | Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝑋 = (𝐵 ∩ 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | ||
Theorem | elin1d 4025 | Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.) |
⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐴) | ||
Theorem | elin2d 4026 | Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.) |
⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐵) | ||
Theorem | elin3 4027 | Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝑋 = ((𝐵 ∩ 𝐶) ∩ 𝐷) ⇒ ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) | ||
Theorem | incom 4028 | Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | ||
Theorem | ineqri 4029* | Inference from membership to intersection. (Contributed by NM, 21-Jun-1993.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∩ 𝐵) = 𝐶 | ||
Theorem | ineq1 4030 | Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) | ||
Theorem | ineq2 4031 | Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) | ||
Theorem | ineq12 4032 | Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) | ||
Theorem | ineq1i 4033 | Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) | ||
Theorem | ineq2i 4034 | Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵) | ||
Theorem | ineq12i 4035 | Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷) | ||
Theorem | ineq1d 4036 | Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) | ||
Theorem | ineq2d 4037 | Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) | ||
Theorem | ineq12d 4038 | Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) | ||
Theorem | ineqan12d 4039 | Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) | ||
Theorem | sseqin2 4040 | A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) | ||
Theorem | nfin 4041 | Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) | ||
Theorem | rabbi2dva 4042* | Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓)) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝜓}) | ||
Theorem | inidm 4043 | Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 ∩ 𝐴) = 𝐴 | ||
Theorem | inass 4044 | Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) |
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) | ||
Theorem | in12 4045 | A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) |
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) | ||
Theorem | in32 4046 | A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵) | ||
Theorem | in13 4047 | A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.) |
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴)) | ||
Theorem | in31 4048 | A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.) |
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴) | ||
Theorem | inrot 4049 | Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.) |
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵) | ||
Theorem | in4 4050 | Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.) |
⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) | ||
Theorem | inindi 4051 | Intersection distributes over itself. (Contributed by NM, 6-May-1994.) |
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) | ||
Theorem | inindir 4052 | Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.) |
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) | ||
Theorem | inss1 4053 | The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.) |
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | ||
Theorem | inss2 4054 | The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.) |
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | ||
Theorem | ssin 4055 | Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) | ||
Theorem | ssini 4056 | An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ 𝐴 ⊆ 𝐶 ⇒ ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶) | ||
Theorem | ssind 4057 | A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∩ 𝐶)) | ||
Theorem | ssrin 4058 | Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) | ||
Theorem | sslin 4059 | Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵)) | ||
Theorem | ssrind 4060 | Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) | ||
Theorem | ss2in 4061 | Intersection of subclasses. (Contributed by NM, 5-May-2000.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷)) | ||
Theorem | ssinss1 4062 | Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.) |
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | ||
Theorem | inss 4063 | Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.) |
⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) | ||
Theorem | rexin 4064 | Restricted existential quantification over intersection. (Contributed by Peter Mazsa, 17-Dec-2018.) |
⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝜑)) | ||
Theorem | dfss7 4065* | Alternate definition of subclass relationship. (Contributed by AV, 1-Aug-2022.) |
⊢ (𝐵 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵) | ||
Syntax | csymdif 4066 | Declare the syntax for symmetric difference. |
class (𝐴 △ 𝐵) | ||
Definition | df-symdif 4067 | Define the symmetric difference of two classes. Alternate definitions are dfsymdif2 4075, dfsymdif3 4119 and dfsymdif4 4073. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) | ||
Theorem | symdifcom 4068 | Symmetric difference commutes. (Contributed by Scott Fenton, 24-Apr-2012.) |
⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴) | ||
Theorem | symdifeq1 4069 | Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.) |
⊢ (𝐴 = 𝐵 → (𝐴 △ 𝐶) = (𝐵 △ 𝐶)) | ||
Theorem | symdifeq2 4070 | Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.) |
⊢ (𝐴 = 𝐵 → (𝐶 △ 𝐴) = (𝐶 △ 𝐵)) | ||
Theorem | nfsymdif 4071 | Hypothesis builder for symmetric difference. (Contributed by Scott Fenton, 19-Feb-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 △ 𝐵) | ||
Theorem | elsymdif 4072 | Membership in a symmetric difference. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)) | ||
Theorem | dfsymdif4 4073* | Alternate definition of the symmetric difference. (Contributed by NM, 17-Aug-2004.) (Revised by AV, 17-Aug-2022.) |
⊢ (𝐴 △ 𝐵) = {𝑥 ∣ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)} | ||
Theorem | elsymdifxor 4074 | Membership in a symmetric difference is an exclusive-or relationship. (Contributed by David A. Wheeler, 26-Apr-2020.) (Proof shortened by BJ, 13-Aug-2022.) |
⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶)) | ||
Theorem | dfsymdif2 4075* | Alternate definition of the symmetric difference. (Contributed by BJ, 30-Apr-2020.) |
⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} | ||
Theorem | symdifass 4076 | Symmetric difference is associative. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by BJ, 7-Sep-2022.) |
⊢ ((𝐴 △ 𝐵) △ 𝐶) = (𝐴 △ (𝐵 △ 𝐶)) | ||
Theorem | symdifassOLD 4077 | Obsolete proof of symdifass 4076 as of 7-Sep-2022. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 △ (𝐵 △ 𝐶)) = ((𝐴 △ 𝐵) △ 𝐶) | ||
Theorem | difsssymdif 4078 | The symmetric difference contains one of the differences. (Proposed by BJ, 18-Aug-2022.) (Contributed by AV, 19-Aug-2022.) |
⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵) | ||
Theorem | difsymssdifssd 4079 | If the symmetric difference is contained in 𝐶, so is one of the differences. (Contributed by AV, 17-Aug-2022.) |
⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶) | ||
Theorem | symdif2OLD 4080* | Obsolete version of dfsymdif4 4073 as of 18-Aug-2022. Two ways to express symmetric difference (df-symdif 4067). (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = {𝑥 ∣ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)} | ||
Theorem | unabs 4081 | Absorption law for union. (Contributed by NM, 16-Apr-2006.) |
⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴 | ||
Theorem | inabs 4082 | Absorption law for intersection. (Contributed by NM, 16-Apr-2006.) |
⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴 | ||
Theorem | nssinpss 4083 | Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴) | ||
Theorem | nsspssun 4084 | Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.) |
⊢ (¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ (𝐴 ∪ 𝐵)) | ||
Theorem | dfss4 4085 | Subclass defined in terms of class difference. See comments under dfun2 4086. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴) | ||
Theorem | dfun2 4086 | An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 4087 and dfss4 4085 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation ∖ (class difference). (Contributed by NM, 10-Jun-2004.) |
⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) | ||
Theorem | dfin2 4087 | An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 4086. Another version is given by dfin4 4094. (Contributed by NM, 10-Jun-2004.) |
⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵)) | ||
Theorem | difin 4088 | Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | ||
Theorem | ssdifim 4089 | Implication of a class difference with a subclass. (Contributed by AV, 3-Jan-2022.) |
⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵)) | ||
Theorem | ssdifsym 4090 | Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.) |
⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) ↔ 𝐴 = (𝑉 ∖ 𝐵))) | ||
Theorem | dfss5 4091* | Alternate definition of subclass relationship: a class 𝐴 is a subclass of another class 𝐵 iff each element of 𝐴 is equal to an element of 𝐵. (Contributed by AV, 13-Nov-2020.) |
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝑦) | ||
Theorem | dfun3 4092 | Union defined in terms of intersection (De Morgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) | ||
Theorem | dfin3 4093 | Intersection defined in terms of union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝐴 ∩ 𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) | ||
Theorem | dfin4 4094 | Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.) |
⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) | ||
Theorem | invdif 4095 | Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.) |
⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) | ||
Theorem | indif 4096 | Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) |
⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵) | ||
Theorem | indif2 4097 | Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.) |
⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶) | ||
Theorem | indif1 4098 | Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶) | ||
Theorem | indifcom 4099 | Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶)) | ||
Theorem | indi 4100 | Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)) |
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