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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sspsstr 4101 | Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | ||
Theorem | psssstr 4102 | Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) | ||
Theorem | psstrd 4103 | Proper subclass inclusion is transitive. Deduction form of psstr 4100. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊊ 𝐵) & ⊢ (𝜑 → 𝐵 ⊊ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) | ||
Theorem | sspsstrd 4104 | Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4101. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ⊊ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) | ||
Theorem | psssstrd 4105 | Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4102. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊊ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) | ||
Theorem | npss 4106 | A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3992. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵)) | ||
Theorem | ssnelpss 4107 | A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.) |
⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) | ||
Theorem | ssnelpssd 4108 | Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4107. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐵) | ||
Theorem | ssexnelpss 4109* | If there is an element of a class which is not contained in a subclass, the subclass is a proper subclass. (Contributed by AV, 29-Jan-2020.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵) | ||
Theorem | dfdif3 4110* | Alternate definition of class difference. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.) |
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦} | ||
Theorem | difeq1 4111 | Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) | ||
Theorem | difeq2 4112 | Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) | ||
Theorem | difeq12 4113 | Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) | ||
Theorem | difeq1i 4114 | Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶) | ||
Theorem | difeq2i 4115 | Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵) | ||
Theorem | difeq12i 4116 | Equality inference for class difference. (Contributed by NM, 29-Aug-2004.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷) | ||
Theorem | difeq1d 4117 | Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) | ||
Theorem | difeq2d 4118 | Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) | ||
Theorem | difeq12d 4119 | Equality deduction for class difference. (Contributed by FL, 29-May-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) | ||
Theorem | difeqri 4120* | Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∖ 𝐵) = 𝐶 | ||
Theorem | nfdif 4121 | Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2129, ax-11 2146, ax-12 2166. (Revised by SN, 14-May-2025.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵) | ||
Theorem | nfdifOLD 4122 | Obsolete version of nfdif 4121 as of 14-May-2025. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵) | ||
Theorem | eldifi 4123 | Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵) | ||
Theorem | eldifn 4124 | Implication of membership in a class difference. (Contributed by NM, 3-May-1994.) |
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → ¬ 𝐴 ∈ 𝐶) | ||
Theorem | elndif 4125 | A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.) |
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) | ||
Theorem | neldif 4126 | Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶) | ||
Theorem | difdif 4127 | Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.) |
⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴 | ||
Theorem | difss 4128 | Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | ||
Theorem | difssd 4129 | A difference of two classes is contained in the minuend. Deduction form of difss 4128. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) | ||
Theorem | difss2 4130 | If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵) | ||
Theorem | difss2d 4131 | If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4130. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | ssdifss 4132 | Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵) | ||
Theorem | ddif 4133 | Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.) |
⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 | ||
Theorem | ssconb 4134 | Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴))) | ||
Theorem | sscon 4135 | Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) | ||
Theorem | ssdif 4136 | Difference law for subsets. (Contributed by NM, 28-May-1998.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) | ||
Theorem | ssdifd 4137 | If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 4136. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) | ||
Theorem | sscond 4138 | If 𝐴 is contained in 𝐵, then (𝐶 ∖ 𝐵) is contained in (𝐶 ∖ 𝐴). Deduction form of sscon 4135. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) | ||
Theorem | ssdifssd 4139 | If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is also contained in 𝐵. Deduction form of ssdifss 4132. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵) | ||
Theorem | ssdif2d 4140 | If 𝐴 is contained in 𝐵 and 𝐶 is contained in 𝐷, then (𝐴 ∖ 𝐷) is contained in (𝐵 ∖ 𝐶). Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ⊆ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶)) | ||
Theorem | raldifb 4141 | Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.) |
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑) | ||
Theorem | rexdifi 4142 | Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.) |
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑) | ||
Theorem | complss 4143 | Complementation reverses inclusion. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 19-Mar-2021.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) | ||
Theorem | compleq 4144 | Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.) |
⊢ (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵)) | ||
Theorem | elun 4145 | Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.) |
⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) | ||
Theorem | elunnel1 4146 | A member of a union that is not a member of the first class, is a member of the second class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶) | ||
Theorem | elunnel2 4147 | A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵) | ||
Theorem | uneqri 4148* | Inference from membership to union. (Contributed by NM, 21-Jun-1993.) |
⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∪ 𝐵) = 𝐶 | ||
Theorem | unidm 4149 | Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝐴 ∪ 𝐴) = 𝐴 | ||
Theorem | uncom 4150 | Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) | ||
Theorem | equncom 4151 | If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4151 was automatically derived from equncomVD 44449 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) |
⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)) | ||
Theorem | equncomi 4152 | Inference form of equncom 4151. equncomi 4152 was automatically derived from equncomiVD 44450 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) |
⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ 𝐴 = (𝐶 ∪ 𝐵) | ||
Theorem | uneq1 4153 | Equality theorem for the union of two classes. (Contributed by NM, 15-Jul-1993.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) | ||
Theorem | uneq2 4154 | Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) | ||
Theorem | uneq12 4155 | Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | ||
Theorem | uneq1i 4156 | Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) | ||
Theorem | uneq2i 4157 | Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵) | ||
Theorem | uneq12i 4158 | Equality inference for the union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷) | ||
Theorem | uneq1d 4159 | Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) | ||
Theorem | uneq2d 4160 | Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) | ||
Theorem | uneq12d 4161 | Equality deduction for the union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | ||
Theorem | nfun 4162 | Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) Avoid ax-10 2129, ax-11 2146, ax-12 2166. (Revised by SN, 14-May-2025.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵) | ||
Theorem | nfunOLD 4163 | Obsolete version of nfun 4162 as of 14-May-2025. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵) | ||
Theorem | unass 4164 | Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) | ||
Theorem | un12 4165 | A rearrangement of union. (Contributed by NM, 12-Aug-2004.) |
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) | ||
Theorem | un23 4166 | A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ 𝐵) | ||
Theorem | un4 4167 | A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.) |
⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) | ||
Theorem | unundi 4168 | Union distributes over itself. (Contributed by NM, 17-Aug-2004.) |
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) | ||
Theorem | unundir 4169 | Union distributes over itself. (Contributed by NM, 17-Aug-2004.) |
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶)) | ||
Theorem | ssun1 4170 | Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | ||
Theorem | ssun2 4171 | Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) | ||
Theorem | ssun3 4172 | Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | ||
Theorem | ssun4 4173 | Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵)) | ||
Theorem | elun1 4174 | Membership law for union of classes. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶)) | ||
Theorem | elun2 4175 | Membership law for union of classes. (Contributed by NM, 30-Aug-1993.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵)) | ||
Theorem | elunant 4176 | A statement is true for every element of the union of a pair of classes if and only if it is true for every element of the first class and for every element of the second class. (Contributed by BTernaryTau, 27-Sep-2023.) |
⊢ ((𝐶 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 → 𝜑) ∧ (𝐶 ∈ 𝐵 → 𝜑))) | ||
Theorem | unss1 4177 | 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 4178 | 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 4179 | 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 4180 | Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) | ||
Theorem | ssequn2 4181 | A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) | ||
Theorem | unss 4182 | 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 4183 | 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 4184 | 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 4185 | If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 4182. Partial converse of unssd 4184. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | unssbd 4186 | If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 4182. Partial converse of unssd 4184. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | ssun 4187 | 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 4188 | Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.) |
⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | ralunb 4189 | 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 4190 | Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑) | ||
Theorem | elini 4191 | Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) | ||
Theorem | elind 4192 | 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 4193 | Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵) | ||
Theorem | elinel2 4194 | Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐶) | ||
Theorem | elin2 4195 | 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 4196 | 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 4197 | 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 4198 | 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 4199 | Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 21-Jun-1993.) (Proof shortened by SN, 12-Dec-2023.) |
⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | ||
Theorem | ineqcom 4200 | Two ways of expressing that two classes have a given intersection. This is often used when that given intersection is the empty set, in which case the statement displays two ways of expressing that two classes are disjoint (when 𝐶 = ∅: ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)). (Contributed by Peter Mazsa, 22-Mar-2017.) |
⊢ ((𝐴 ∩ 𝐵) = 𝐶 ↔ (𝐵 ∩ 𝐴) = 𝐶) |
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