HomeHome Metamath Proof Explorer
Theorem List (p. 44 of 463)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-29030)
  Hilbert Space Explorer  Hilbert Space Explorer
(29031-30553)
  Users' Mathboxes  Users' Mathboxes
(30554-46225)
 

Theorem List for Metamath Proof Explorer - 4301-4400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdifid 4301 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.) (Revised by David Abernethy, 17-Jun-2012.)
(𝐴𝐴) = ∅
 
TheoremdifidALT 4302 Alternate proof of difid 4301. Shorter, but requiring ax-8 2114, df-clel 2818. (Contributed by NM, 22-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐴) = ∅
 
Theoremdif0 4303 The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ ∅) = 𝐴
 
Theoremab0w 4304* The class of sets verifying a property is the empty class if and only if that property is a contradiction. Version of ab0 4305 using implicit substitution, which requires fewer axioms. (Contributed by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       ({𝑥𝜑} = ∅ ↔ ∀𝑦 ¬ 𝜓)
 
Theoremab0 4305 The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 4311 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne 2944). (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2818, ax-8 2114. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by SN, 8-Sep-2024.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
Theoremab0OLD 4306 Obsolete version of ab0 4305 as of 8-Sep-2024. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2818, ax-8 2114. (Revised by Gino Giotto, 30-Aug-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
Theoremab0ALT 4307 Alternate proof of ab0 4305, shorter but using more axioms. (Contributed by BJ, 19-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
Theoremdfnf5 4308 Characterization of nonfreeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.)
(Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
 
Theoremab0orv 4309* The class abstraction defined by a formula not containing the abstraction variable is either the empty set or the universal class. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.) Reduce axiom usage. (Revised by Gino Giotto, 30-Aug-2024.)
({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅)
 
Theoremab0orvALT 4310* Alternate proof of ab0orv 4309, shorter but using more axioms. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅)
 
Theoremabn0 4311 Nonempty class abstraction. See also ab0 4305. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) Avoid df-clel 2818, ax-8 2114. (Revised by Gino Giotto, 30-Aug-2024.)
({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
 
Theoremabn0OLD 4312 Obsolete version of abn0 4311 as of 30-Aug-2024. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
 
Theoremrab0 4313 Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
{𝑥 ∈ ∅ ∣ 𝜑} = ∅
 
Theoremrabeq0w 4314* Condition for a restricted class abstraction to be empty. Version of rabeq0 4315 using implicit substitution, which does not require ax-10 2143, ax-11 2160, ax-12 2177, but requires ax-8 2114. (Contributed by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑦𝐴 ¬ 𝜓)
 
Theoremrabeq0 4315 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.) (Revised by BJ, 16-Jul-2021.)
({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥𝐴 ¬ 𝜑)
 
Theoremrabn0 4316 Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.) (Revised by BJ, 16-Jul-2021.)
({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
 
Theoremrabxm 4317* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
 
Theoremrabnc 4318 Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅
 
Theoremelneldisj 4319* The set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element are disjoint. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.)
𝐸 = {𝑠𝐴𝐵𝐶}    &   𝑁 = {𝑠𝐴𝐵𝐶}       (𝐸𝑁) = ∅
 
Theoremelnelun 4320* The union of the set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element yields the original set. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.)
𝐸 = {𝑠𝐴𝐵𝐶}    &   𝑁 = {𝑠𝐴𝐵𝐶}       (𝐸𝑁) = 𝐴
 
Theoremun0 4321 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 15-Jul-1993.)
(𝐴 ∪ ∅) = 𝐴
 
Theoremin0 4322 The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 21-Jun-1993.)
(𝐴 ∩ ∅) = ∅
 
Theorem0un 4323 The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∪ 𝐴) = 𝐴
 
Theorem0in 4324 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∩ 𝐴) = ∅
 
Theoreminv1 4325 The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
(𝐴 ∩ V) = 𝐴
 
Theoremunv 4326 The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
(𝐴 ∪ V) = V
 
Theorem0ss 4327 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.)
∅ ⊆ 𝐴
 
Theoremss0b 4328 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
(𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
 
Theoremss0 4329 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
(𝐴 ⊆ ∅ → 𝐴 = ∅)
 
Theoremsseq0 4330 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)
 
Theoremssn0 4331 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
((𝐴𝐵𝐴 ≠ ∅) → 𝐵 ≠ ∅)
 
Theorem0dif 4332 The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
(∅ ∖ 𝐴) = ∅
 
Theoremabf 4333 A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2114, ax-10 2143, ax-11 2160, ax-12 2177. (Revised by Gino Giotto, 30-Jun-2024.)
¬ 𝜑       {𝑥𝜑} = ∅
 
TheoremabfOLD 4334 Obsolete version of abf 4333 as of 28-Jun-2024. (Contributed by NM, 20-Jan-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ 𝜑       {𝑥𝜑} = ∅
 
Theoremeq0rdv 4335* Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.) Avoid ax-8 2114, df-clel 2818. (Revised by Gino Giotto, 6-Sep-2024.)
(𝜑 → ¬ 𝑥𝐴)       (𝜑𝐴 = ∅)
 
Theoremeq0rdvALT 4336* Alternate proof of eq0rdv 4335. Shorter, but requiring df-clel 2818, ax-8 2114. (Contributed by NM, 11-Jul-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ¬ 𝑥𝐴)       (𝜑𝐴 = ∅)
 
Theoremcsbprc 4337 The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.) (Proof shortened by JJ, 27-Aug-2021.)
𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
 
Theoremcsb0 4338 The proper substitution of a class into the empty set is the empty set. (Contributed by NM, 18-Aug-2018.)
𝐴 / 𝑥∅ = ∅
 
Theoremsbcel12 4339 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 
Theoremsbceqg 4340 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
 
Theoremsbceqi 4341 Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V    &   𝐴 / 𝑥𝐵 = 𝐷    &   𝐴 / 𝑥𝐶 = 𝐸       ([𝐴 / 𝑥]𝐵 = 𝐶𝐷 = 𝐸)
 
Theoremsbcnel12g 4342 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
Theoremsbcne12 4343 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 
Theoremsbcel1g 4344* Move proper substitution in and out of a membership relation. Note that the scope of [𝐴 / 𝑥] is the wff 𝐵𝐶, whereas the scope of 𝐴 / 𝑥 is the class 𝐵. (Contributed by NM, 10-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐶))
 
Theoremsbceq1g 4345* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
 
Theoremsbcel2 4346* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶)
 
Theoremsbceq2g 4347* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐵 = 𝐴 / 𝑥𝐶))
 
Theoremcsbcom 4348* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
 
Theoremsbcnestgfw 4349* Nest the composition of two substitutions. Version of sbcnestgf 4354 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by Mario Carneiro, 11-Nov-2016.) (Revised by Gino Giotto, 26-Jan-2024.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 
Theoremcsbnestgfw 4350* Nest the composition of two substitutions. Version of csbnestgf 4355 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 26-Jan-2024.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 
Theoremsbcnestgw 4351* Nest the composition of two substitutions. Version of sbcnestg 4356 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 27-Nov-2005.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 
Theoremcsbnestgw 4352* Nest the composition of two substitutions. Version of csbnestg 4357 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 
Theoremsbcco3gw 4353* Composition of two substitutions. Version of sbcco3g 4358 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 27-Nov-2005.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
 
Theoremsbcnestgf 4354 Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker sbcnestgfw 4349 when possible. (Contributed by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 
Theoremcsbnestgf 4355 Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker csbnestgfw 4350 when possible. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) (New usage is discouraged.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 
Theoremsbcnestg 4356* Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker sbcnestgw 4351 when possible. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 
Theoremcsbnestg 4357* Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker csbnestgw 4352 when possible. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) (New usage is discouraged.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 
Theoremsbcco3g 4358* Composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker sbcco3gw 4353 when possible. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
 
Theoremcsbco3g 4359* Composition of two class substitutions. Usage of this theorem is discouraged because it depends on ax-13 2373. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
 
Theoremcsbnest1g 4360 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)
 
Theoremcsbidm 4361* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) (Revised by NM, 18-Aug-2018.)
𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
 
Theoremcsbvarg 4362 The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥𝑥 = 𝐴)
 
Theoremcsbvargi 4363 The proper substitution of a class for a setvar variable results in the class (if the class exists), in inference form of csbvarg 4362. (Contributed by Giovanni Mascellani, 30-May-2019.)
𝐴 ∈ V       𝐴 / 𝑥𝑥 = 𝐴
 
Theoremsbccsb 4364* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝜑𝑦𝐴 / 𝑥{𝑦𝜑})
 
Theoremsbccsb2 4365 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑})
 
Theoremrspcsbela 4366* Special case related to rspsbc 3808. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
 
Theoremsbnfc2 4367* Two ways of expressing "𝑥 is (effectively) not free in 𝐴". (Contributed by Mario Carneiro, 14-Oct-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
 
Theoremcsbab 4368* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 19-Aug-2018.)
𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}
 
Theoremcsbun 4369 Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) (Revised by NM, 13-Sep-2018.)
𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 
Theoremcsbin 4370 Distribute proper substitution into a class through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) (Revised by NM, 18-Aug-2018.)
𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 
Theoremcsbie2df 4371* Conversion of implicit substitution to explicit class substitution. This version of csbiedf 3859 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 by substituting twice. Deduction form of csbie2 3870. (Contributed by AV, 29-Mar-2024.)
𝑥𝜑    &   (𝜑𝑥𝐶)    &   (𝜑𝑥𝐷)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)    &   ((𝜑𝑦 = 𝐴) → 𝐶 = 𝐷)       (𝜑𝐴 / 𝑥𝐵 = 𝐷)
 
Theorem2nreu 4372* If there are two different sets fulfilling a wff (by implicit substitution), then there is no unique set fulfilling the wff. (Contributed by AV, 20-Jun-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝑋𝐵𝑋𝐴𝐵) → ((𝜓𝜒) → ¬ ∃!𝑥𝑋 𝜑))
 
Theoremun00 4373 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)
 
Theoremvss 4374 Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(V ⊆ 𝐴𝐴 = V)
 
Theorem0pss 4375 The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.)
(∅ ⊊ 𝐴𝐴 ≠ ∅)
 
Theoremnpss0 4376 No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
¬ 𝐴 ⊊ ∅
 
Theorempssv 4377 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
(𝐴 ⊊ V ↔ ¬ 𝐴 = V)
 
Theoremdisj 4378* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) Avoid ax-10 2143, ax-11 2160, ax-12 2177. (Revised by Gino Giotto, 28-Jun-2024.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
 
TheoremdisjOLD 4379* Obsolete version of disj 4378 as of 28-Jun-2024. (Contributed by NM, 17-Feb-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
 
Theoremdisjr 4380* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
 
Theoremdisj1 4381* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
 
Theoremreldisj 4382 Two ways of saying that two classes are disjoint, using the complement of 𝐵 relative to a universe 𝐶. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid ax-12 2177. (Revised by Gino Giotto, 28-Jun-2024.)
(𝐴𝐶 → ((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶𝐵)))
 
TheoremreldisjOLD 4383 Obsolete version of reldisj 4382 as of 28-Jun-2024. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐶 → ((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶𝐵)))
 
Theoremdisj3 4384 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
 
Theoremdisjne 4385 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)
 
Theoremdisjeq0 4386 Two disjoint sets are equal iff both are empty. (Contributed by AV, 19-Jun-2022.)
((𝐴𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
 
Theoremdisjel 4387 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)
 
Theoremdisj2 4388 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))
 
Theoremdisj4 4389 Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)
((𝐴𝐵) = ∅ ↔ ¬ (𝐴𝐵) ⊊ 𝐴)
 
Theoremssdisj 4390 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.) (Proof shortened by JJ, 14-Jul-2021.)
((𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)
 
Theoremdisjpss 4391 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
(((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴𝐵))
 
Theoremundisj1 4392 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
(((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐵) ∩ 𝐶) = ∅)
 
Theoremundisj2 4393 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
(((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)
 
Theoremssindif0 4394 Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
(𝐴𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)
 
Theoreminelcm 4395 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
((𝐴𝐵𝐴𝐶) → (𝐵𝐶) ≠ ∅)
 
Theoremminel 4396 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.) (Proof shortened by JJ, 14-Jul-2021.)
((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)
 
Theoremundif4 4397 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))
 
Theoremdisjssun 4398 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐶))
 
Theoremvdif0 4399 Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.)
(𝐴 = V ↔ (V ∖ 𝐴) = ∅)
 
Theoremdifrab0eq 4400* If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
((𝑉 ∖ {𝑥𝑉𝜑}) = ∅ ↔ 𝑉 = {𝑥𝑉𝜑})
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46225
  Copyright terms: Public domain < Previous  Next >