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Theorem List for Metamath Proof Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrabsnifsb 4201* A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)
{𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)

Theoremrabsnif 4202* A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)
(𝑥 = 𝐴 → (𝜑𝜓))       {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)

Theoremrabrsn 4203* A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Proof shortened by AV, 21-Jul-2019.)
(𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))

Theoremeuabsn2 4204* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})

Theoremeuabsn 4205 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
(∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})

Theoremreusn 4206* A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
(∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})

Theoremabsneu 4207 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)

Theoremrabsneu 4208 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)

Theoremeusn 4209* Two ways to express "𝐴 is a singleton." (Contributed by NM, 30-Oct-2010.)
(∃!𝑥 𝑥𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Theoremrabsnt 4210* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
𝐵 ∈ V    &   (𝑥 = 𝐵 → (𝜑𝜓))       ({𝑥𝐴𝜑} = {𝐵} → 𝜓)

Theoremprcom 4211 Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993.)
{𝐴, 𝐵} = {𝐵, 𝐴}

Theorempreq1 4212 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
(𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})

Theorempreq2 4213 Equality theorem for unordered pairs. (Contributed by NM, 15-Jul-1993.)
(𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})

Theorempreq12 4214 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})

Theorempreq1i 4215 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵       {𝐴, 𝐶} = {𝐵, 𝐶}

Theorempreq2i 4216 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵       {𝐶, 𝐴} = {𝐶, 𝐵}

Theorempreq12i 4217 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       {𝐴, 𝐶} = {𝐵, 𝐷}

Theorempreq1d 4218 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})

Theorempreq2d 4219 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵})

Theorempreq12d 4220 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})

Theoremtpeq1 4221 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})

Theoremtpeq2 4222 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})

Theoremtpeq3 4223 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})

Theoremtpeq1d 4224 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})

Theoremtpeq2d 4225 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})

Theoremtpeq3d 4226 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})

Theoremtpeq123d 4227 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)    &   (𝜑𝐸 = 𝐹)       (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})

Theoremtprot 4228 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
{𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}

Theoremtpcoma 4229 Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015.)
{𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}

Theoremtpcomb 4230 Swap 2nd and 3rd members of an unordered triple. (Contributed by NM, 22-May-2015.)
{𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}

Theoremtpass 4231 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
{𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})

Theoremqdass 4232 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷})

Theoremqdassr 4233 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷})

Theoremtpidm12 4234 Unordered triple {𝐴, 𝐴, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
{𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}

Theoremtpidm13 4235 Unordered triple {𝐴, 𝐵, 𝐴} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
{𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}

Theoremtpidm23 4236 Unordered triple {𝐴, 𝐵, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
{𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}

Theoremtpidm 4237 Unordered triple {𝐴, 𝐴, 𝐴} is just an overlong way to write {𝐴}. (Contributed by David A. Wheeler, 10-May-2015.)
{𝐴, 𝐴, 𝐴} = {𝐴}

Theoremtppreq3 4238 An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
(𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Theoremprid1g 4239 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
(𝐴𝑉𝐴 ∈ {𝐴, 𝐵})

Theoremprid2g 4240 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
(𝐵𝑉𝐵 ∈ {𝐴, 𝐵})

Theoremprid1 4241 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 24-Jun-1993.)
𝐴 ∈ V       𝐴 ∈ {𝐴, 𝐵}

Theoremprid2 4242 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
𝐵 ∈ V       𝐵 ∈ {𝐴, 𝐵}

Theoremprprc1 4243 A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.)
𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})

Theoremprprc2 4244 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

Theoremprprc 4245 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)

Theoremtpid1 4246 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐴 ∈ V       𝐴 ∈ {𝐴, 𝐵, 𝐶}

Theoremtpid2 4247 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐵 ∈ V       𝐵 ∈ {𝐴, 𝐵, 𝐶}

Theoremtpid3g 4248 Closed theorem form of tpid3 4250. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 30-Apr-2021.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Theoremtpid3gOLD 4249 Obsolete proof of tpid3g 4248 as of 30-Apr-2021. Closed theorem form of tpid3 4250. This proof was automatically generated from the virtual deduction proof tpid3gVD 38099 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Theoremtpid3 4250 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 30-Apr-2021.)
𝐶 ∈ V       𝐶 ∈ {𝐴, 𝐵, 𝐶}

Theoremsnnzg 4251 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
(𝐴𝑉 → {𝐴} ≠ ∅)

Theoremsnnz 4252 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
𝐴 ∈ V       {𝐴} ≠ ∅

Theoremprnz 4253 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
𝐴 ∈ V       {𝐴, 𝐵} ≠ ∅

Theoremprnzg 4254 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.) (Proof shortened by JJ, 23-Jul-2021.)
(𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)

TheoremprnzgOLD 4255 Obsolete proof of prnzg 4254 as of 23-Jul-2021. (Contributed by FL, 19-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)

Theoremtpnz 4256 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
𝐴 ∈ V       {𝐴, 𝐵, 𝐶} ≠ ∅

Theoremtpnzd 4257 A triplet containing a set is not empty. (Contributed by Thierry Arnoux, 8-Apr-2019.)
(𝜑𝐴𝑉)       (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)

Theoremraltpd 4258* Convert a quantification over a triple to a conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)
((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜃))    &   ((𝜑𝑥 = 𝐶) → (𝜓𝜏))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)       (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜒𝜃𝜏)))

Theoremsnss 4259 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.)
𝐴 ∈ V       (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)

Theoremeldifsn 4260 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
(𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴𝐵𝐴𝐶))

Theoremeldifsni 4261 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
(𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴𝐶)

Theoremneldifsn 4262 The class 𝐴 is not in (𝐵 ∖ {𝐴}). (Contributed by David Moews, 1-May-2017.)
¬ 𝐴 ∈ (𝐵 ∖ {𝐴})

Theoremneldifsnd 4263 The class 𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Theoremrexdifsn 4264 Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
(∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))

Theoremraldifsni 4265 Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.)
(∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵))

Theoremraldifsnb 4266* Restricted universal quantification on a class difference with a singleton in terms of an implication. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
(∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)

Theoremeldifvsn 4267 A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.)
(𝐴𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴𝐵))

Theoremsnssg 4268 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)
(𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))

Theoremdifsn 4269 An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)

Theoremdifprsnss 4270 Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵}

Theoremdifprsn1 4271 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Theoremdifprsn2 4272 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
(𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})

Theoremdiftpsn3 4273 Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Proof shortened by JJ, 23-Jul-2021.)
((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})

Theoremdiftpsn3OLD 4274 Obsolete proof of diftpsn3 4273 as of 23-Jul-2021. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})

Theoremdifpr 4275 Removing two elements as pair of elements corresponds to removing each of the two elements as singletons. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
(𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})

Theoremtpprceq3 4276 An unordered triple is an unordered pair if one of its elements is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
(¬ (𝐶 ∈ V ∧ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Theoremtppreqb 4277 An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)
(¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) ↔ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Theoremdifsnb 4278 (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4269. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Theoremdifsnpss 4279 (𝐵 ∖ {𝐴}) is a proper subclass of 𝐵 if and only if 𝐴 is a member of 𝐵. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)

Theoremsnssi 4280 The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)

Theoremsnssd 4281 The singleton of an element of a class is a subset of the class (deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)       (𝜑 → {𝐴} ⊆ 𝐵)

Theoremdifsnid 4282 If we remove a single element from a class then put it back in, we end up with the original class. (Contributed by NM, 2-Oct-2006.)
(𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Theorempw0 4283 Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝒫 ∅ = {∅}

Theorempwpw0 4284 Compute the power set of the power set of the empty set. (See pw0 4283 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4366, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)
𝒫 {∅} = {∅, {∅}}

Theoremsnsspr1 4285 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
{𝐴} ⊆ {𝐴, 𝐵}

Theoremsnsspr2 4286 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
{𝐵} ⊆ {𝐴, 𝐵}

Theoremsnsstp1 4287 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐴} ⊆ {𝐴, 𝐵, 𝐶}

Theoremsnsstp2 4288 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Theoremsnsstp3 4289 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Theoremprssg 4290 A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))

Theoremprss 4291 A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 23-Jul-2021.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

TheoremprssOLD 4292 Obsolete proof of prss 4291 as of 23-Jul-2021. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

Theoremprssi 4293 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)

Theoremprssd 4294 Deduction version of prssi 4293: A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)

Theoremprsspwg 4295 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)
((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Theoremssprss 4296 A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.)
((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))

Theoremssprsseq 4297 A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))

Theoremsssn 4298 The subsets of a singleton. (Contributed by NM, 24-Apr-2004.)
(𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))

Theoremssunsn2 4299 The property of being sandwiched between two sets naturally splits under union with a singleton. This is the induction hypothesis for the determination of large powersets such as pwtp 4369. (Contributed by Mario Carneiro, 2-Jul-2016.)
((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))

Theoremssunsn 4300 Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)
((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})))

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