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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremifbothda 4101 A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))    &   ((𝜂𝜑) → 𝜓)    &   ((𝜂 ∧ ¬ 𝜑) → 𝜒)       (𝜂𝜃)
 
Theoremifboth 4102 A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))       ((𝜓𝜒) → 𝜃)
 
Theoremifid 4103 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
if(𝜑, 𝐴, 𝐴) = 𝐴
 
Theoremeqif 4104 Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
(𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)))
 
Theoremifval 4105 Another expression of the value of the if predicate, analogous to eqif 4104. See also the more specialized iftrue 4070 and iffalse 4073. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∧ (¬ 𝜑𝐴 = 𝐶)))
 
Theoremelif 4106 Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
(𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴𝐵) ∨ (¬ 𝜑𝐴𝐶)))
 
Theoremifel 4107 Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
(if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑𝐴𝐶) ∨ (¬ 𝜑𝐵𝐶)))
 
Theoremifcl 4108 Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
((𝐴𝐶𝐵𝐶) → if(𝜑, 𝐴, 𝐵) ∈ 𝐶)
 
Theoremifcld 4109 Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
 
Theoremifeqor 4110 The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
 
Theoremifnot 4111 Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)
 
Theoremifan 4112 Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵)
 
Theoremifor 4113 Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵))
 
Theorem2if2 4114 Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)
((𝜑𝜓) → 𝐷 = 𝐴)    &   ((𝜑 ∧ ¬ 𝜓𝜃) → 𝐷 = 𝐵)    &   ((𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜃) → 𝐷 = 𝐶)       (𝜑𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))
 
Theoremifcomnan 4115 Commute the conditions in two nested conditionals if both conditions are not simultaneously true. (Contributed by SO, 15-Jul-2018.)
(¬ (𝜑𝜓) → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶)))
 
Theoremcsbif 4116 Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.)
𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶)
 
Theoremdedth 4117 Weak deduction theorem that eliminates a hypothesis 𝜑, making it become an antecedent. We assume that a proof exists for 𝜑 when the class variable 𝐴 is replaced with a specific class 𝐵. The hypothesis 𝜒 should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 4124. If the inference has other hypotheses with class variable 𝐴, these can be kept by assigning keephyp 4130 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))    &   𝜒       (𝜑𝜓)
 
Theoremdedth2h 4118 Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 4121 but requires that each hypothesis has exactly one class variable. See also comments in dedth 4117. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜒𝜃))    &   (𝐵 = if(𝜓, 𝐵, 𝐷) → (𝜃𝜏))    &   𝜏       ((𝜑𝜓) → 𝜒)
 
Theoremdedth3h 4119 Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4118. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃𝜏))    &   (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏𝜂))    &   (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂𝜁))    &   𝜁       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremdedth4h 4120 Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4118. (Contributed by NM, 16-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏𝜂))    &   (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂𝜁))    &   (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁𝜎))    &   (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎𝜌))    &   𝜌       (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)
 
Theoremdedth2v 4121 Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 4118 is simpler to use. See also comments in dedth 4117. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))    &   𝜃       (𝜑𝜓)
 
Theoremdedth3v 4122 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4121. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   𝜏       (𝜑𝜓)
 
Theoremdedth4v 4123 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4121. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
(𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃𝜏))    &   (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏𝜂))    &   𝜂       (𝜑𝜓)
 
Theoremelimhyp 4124 Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4117. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑𝜓))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜓))    &   𝜒       𝜓
 
Theoremelimhyp2v 4125 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜑𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏𝜂))    &   (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂𝜃))    &   𝜏       𝜃
 
Theoremelimhyp3v 4126 Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))    &   (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))    &   (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜏))    &   𝜂       𝜏
 
Theoremelimhyp4v 4127 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4117). (Contributed by NM, 16-Apr-2005.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   (𝐹 = if(𝜑, 𝐹, 𝐺) → (𝜏𝜓))    &   (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))    &   (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))    &   (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜌))    &   (𝐺 = if(𝜑, 𝐹, 𝐺) → (𝜌𝜓))    &   𝜂       𝜓
 
Theoremelimel 4128 Eliminate a membership hypothesis for weak deduction theorem, when special case 𝐵𝐶 is provable. (Contributed by NM, 15-May-1999.)
𝐵𝐶       if(𝐴𝐶, 𝐴, 𝐵) ∈ 𝐶
 
Theoremelimdhyp 4129 Version of elimhyp 4124 where the hypothesis is deduced from the final antecedent. See divalg 15069 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
(𝜑𝜓)    &   (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃𝜒))    &   𝜃       𝜒
 
Theoremkeephyp 4130 Transform a hypothesis 𝜓 that we want to keep (but contains the same class variable 𝐴 used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))    &   𝜓    &   𝜒       𝜃
 
Theoremkeephyp2v 4131 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4117). (Contributed by NM, 16-Apr-2005.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏𝜂))    &   (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂𝜃))    &   𝜓    &   𝜏       𝜃
 
Theoremkeephyp3v 4132 Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜌𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))    &   (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))    &   (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜏))    &   𝜌    &   𝜂       𝜏
 
Theoremkeepel 4133 Keep a membership hypothesis for weak deduction theorem, when special case 𝐵𝐶 is provable. (Contributed by NM, 14-Aug-1999.)
𝐴𝐶    &   𝐵𝐶       if(𝜑, 𝐴, 𝐵) ∈ 𝐶
 
Theoremifex 4134 Conditional operator existence. (Contributed by NM, 2-Sep-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       if(𝜑, 𝐴, 𝐵) ∈ V
 
Theoremifexg 4135 Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
 
2.1.16  Power classes
 
Syntaxcpw 4136 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
class 𝒫 𝐴
 
Theorempwjust 4137* Soundness justification theorem for df-pw 4138. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥𝐴} = {𝑦𝑦𝐴}
 
Definitiondf-pw 4138* Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if 𝐴 = {3, 5, 7}, then 𝒫 𝐴 = {∅, {3}, {5}, {7}, {3, 5}, {3, 7}, {5, 7}, {3, 5, 7}} (ex-pw 27174). We will later introduce the Axiom of Power Sets ax-pow 4813, which can be expressed in class notation per pwexg 4820. Still later we will prove, in hashpw 13179, that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 24-Jun-1993.)
𝒫 𝐴 = {𝑥𝑥𝐴}
 
Theorempweq 4139 Equality theorem for power class. (Contributed by NM, 21-Jun-1993.)
(𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
 
Theorempweqi 4140 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
𝐴 = 𝐵       𝒫 𝐴 = 𝒫 𝐵
 
Theorempweqd 4141 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → 𝒫 𝐴 = 𝒫 𝐵)
 
Theoremelpw 4142 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
𝐴 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 
Theoremselpw 4143* Setvar variable membership in a power class (common case). See elpw 4142. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝑥 ∈ 𝒫 𝐴𝑥𝐴)
 
Theoremelpwg 4144 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4797. (Contributed by NM, 6-Aug-2000.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 
Theoremelpwd 4145 Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)
 
Theoremelpwi 4146 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
(𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 
Theoremelpwb 4147 Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
(𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
 
Theoremelpwid 4148 An element of a power class is a subclass. Deduction form of elpwi 4146. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ 𝒫 𝐵)       (𝜑𝐴𝐵)
 
Theoremelelpwi 4149 If 𝐴 belongs to a part of 𝐶 then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)
 
Theoremnfpw 4150 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝐴       𝑥𝒫 𝐴
 
Theorempwidg 4151 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐴𝑉𝐴 ∈ 𝒫 𝐴)
 
Theorempwid 4152 A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V       𝐴 ∈ 𝒫 𝐴
 
Theorempwss 4153* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
(𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
2.1.17  Unordered and ordered pairs
 
Theoremsnjust 4154* Soundness justification theorem for df-sn 4156. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}
 
Syntaxcsn 4155 Extend class notation to include singleton.
class {𝐴}
 
Definitiondf-sn 4156* Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of V, although it is not very meaningful in this case. For an alternate definition see dfsn2 4168. (Contributed by NM, 21-Jun-1993.)
{𝐴} = {𝑥𝑥 = 𝐴}
 
Syntaxcpr 4157 Extend class notation to include unordered pair.
class {𝐴, 𝐵}
 
Definitiondf-pr 4158 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, 𝐴 ∈ {1, -1} → (𝐴↑2) = 1 (ex-pr 27175). They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 4244. For a more traditional definition, but requiring a dummy variable, see dfpr2 4173. (Contributed by NM, 21-Jun-1993.)
{𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
 
Syntaxctp 4159 Extend class notation to include unordered triplet.
class {𝐴, 𝐵, 𝐶}
 
Definitiondf-tp 4160 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
{𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
 
Syntaxcop 4161 Extend class notation to include ordered pair.
class 𝐴, 𝐵
 
Definitiondf-op 4162* Definition of an ordered pair, equivalent to Kuratowski's definition {{𝐴}, {𝐴, 𝐵}} when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 4400, opprc2 4401, and 0nelop 4930). For Kuratowski's actual definition when the arguments are sets, see dfop 4376. For the justifying theorem (for sets) see opth 4915. See dfopif 4374 for an equivalent formulation using the if operation.

Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}, which has different behavior from our df-op 4162 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4162 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses.

There are other ways to define ordered pairs. The basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition 𝐴, 𝐵_2 = {{{𝐴}, ∅}, {{𝐵}}}, justified by opthwiener 4946. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition 𝐴, 𝐵_3 = {𝐴, {𝐴, 𝐵}} is justified by opthreg 8475, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is 𝐴, 𝐵_4 = ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})), justified by opthprc 5137. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 13013. An ordered pair of real numbers can also be represented by a complex number as shown by cru 10972. Kuratowski's ordered pair definition is standard for ZFC set theory, but it is very inconvenient to use in New Foundations theory because it is not type-level; a common alternate definition in New Foundations is the definition from [Rosser] p. 281.

Since there are other ways to define ordered pairs, we discourage direct use of this definition so that most theorems won't depend on this particular construction; theorems will instead rely on dfopif 4374. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)

𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
 
Syntaxcotp 4163 Extend class notation to include ordered triple.
class 𝐴, 𝐵, 𝐶
 
Definitiondf-ot 4164 Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)
𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
 
Theoremsneq 4165 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
(𝐴 = 𝐵 → {𝐴} = {𝐵})
 
Theoremsneqi 4166 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
𝐴 = 𝐵       {𝐴} = {𝐵}
 
Theoremsneqd 4167 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴} = {𝐵})
 
Theoremdfsn2 4168 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴} = {𝐴, 𝐴}
 
Theoremelsng 4169 There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
Theoremelsn 4170 There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
𝐴 ∈ V       (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
 
Theoremvelsn 4171 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
(𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
 
Theoremelsni 4172 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
 
Theoremdfpr2 4173* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
 
Theoremelprg 4174 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
 
Theoremelpri 4175 If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
(𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
 
Theoremelpr 4176 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
𝐴 ∈ V       (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
 
Theoremelpr2 4177 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.) (Proof shortened by JJ, 23-Jul-2021.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
 
Theoremelpr2OLD 4178 Obsolete proof of elpr2 4177 as of 23-Jul-2021. (Contributed by NM, 14-Oct-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
 
Theoremnelpri 4179 If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
𝐴𝐵    &   𝐴𝐶        ¬ 𝐴 ∈ {𝐵, 𝐶}
 
Theoremprneli 4180 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using . (Contributed by David A. Wheeler, 10-May-2015.)
𝐴𝐵    &   𝐴𝐶       𝐴 ∉ {𝐵, 𝐶}
 
Theoremnelprd 4181 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)       (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
 
Theoremeldifpr 4182 Membership in a set with two elements removed. Similar to eldifsn 4294 and eldiftp 4206. (Contributed by Mario Carneiro, 18-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
 
Theoremrexdifpr 4183 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
(∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))
 
Theoremsnidg 4184 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
(𝐴𝑉𝐴 ∈ {𝐴})
 
Theoremsnidb 4185 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
(𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
 
Theoremsnid 4186 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
𝐴 ∈ V       𝐴 ∈ {𝐴}
 
Theoremvsnid 4187 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
𝑥 ∈ {𝑥}
 
Theoremelsn2g 4188 There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 28-Oct-2003.)
(𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
Theoremelsn2 4189 There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 12-Jun-1994.)
𝐵 ∈ V       (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
 
Theoremnelsn 4190 If a class is not equal to the class in a singleton, then it is not in the singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by BJ, 4-May-2021.)
(𝐴𝐵 → ¬ 𝐴 ∈ {𝐵})
 
TheoremnelsnOLD 4191 Obsolete proof of nelsn 4190 as of 4-May-2021. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ¬ 𝐴 ∈ {𝐵})
 
Theoremrabeqsn 4192* Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.)
({𝑥𝑉𝜑} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))
 
Theoremrabsssn 4193* Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.)
({𝑥𝑉𝜑} ⊆ {𝑋} ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
 
Theoremralsnsg 4194* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
 
Theoremrexsns 4195* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
(∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
 
Theoremralsng 4196* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
 
Theoremrexsng 4197* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
 
Theorem2ralsng 4198* Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑𝜒))
 
Theoremexsnrex 4199 There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)
(∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥𝑀 𝑀 = {𝑥})
 
Theoremralsn 4200* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥 ∈ {𝐴}𝜑𝜓)
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