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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | difid 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.) |
⊢ (𝐴 ∖ 𝐴) = ∅ | ||
Theorem | difidALT 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.) |
⊢ (𝐴 ∖ 𝐴) = ∅ | ||
Theorem | dif0 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 | ||
Theorem | ab0w 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝜓) | ||
Theorem | ab0 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.) |
⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) | ||
Theorem | ab0OLD 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.) |
⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) | ||
Theorem | ab0ALT 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.) |
⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) | ||
Theorem | dfnf5 4308 | Characterization of nonfreeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.) |
⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)) | ||
Theorem | ab0orv 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.) |
⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅) | ||
Theorem | ab0orvALT 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.) |
⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅) | ||
Theorem | abn0 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.) |
⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) | ||
Theorem | abn0OLD 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.) |
⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) | ||
Theorem | rab0 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} = ∅ | ||
Theorem | rabeq0w 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓) | ||
Theorem | rabeq0 4315 | Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.) (Revised by BJ, 16-Jul-2021.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | ||
Theorem | rabn0 4316 | Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.) (Revised by BJ, 16-Jul-2021.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rabxm 4317* | Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) |
⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) | ||
Theorem | rabnc 4318 | Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅ | ||
Theorem | elneldisj 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 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} & ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} ⇒ ⊢ (𝐸 ∩ 𝑁) = ∅ | ||
Theorem | elnelun 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 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} & ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} ⇒ ⊢ (𝐸 ∪ 𝑁) = 𝐴 | ||
Theorem | un0 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 | ||
Theorem | in0 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 ∩ ∅) = ∅ | ||
Theorem | 0un 4323 | The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (∅ ∪ 𝐴) = 𝐴 | ||
Theorem | 0in 4324 | The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (∅ ∩ 𝐴) = ∅ | ||
Theorem | inv1 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 | ||
Theorem | unv 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 | ||
Theorem | 0ss 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 | ||
Theorem | ss0b 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 = ∅) | ||
Theorem | ss0 4329 | Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.) |
⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | ||
Theorem | sseq0 4330 | A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅) | ||
Theorem | ssn0 4331 | A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐵 ≠ ∅) | ||
Theorem | 0dif 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) = ∅ | ||
Theorem | abf 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 ⇒ ⊢ {𝑥 ∣ 𝜑} = ∅ | ||
Theorem | abfOLD 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 ⇒ ⊢ {𝑥 ∣ 𝜑} = ∅ | ||
Theorem | eq0rdv 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 → ¬ 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = ∅) | ||
Theorem | eq0rdvALT 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 → ¬ 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = ∅) | ||
Theorem | csbprc 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 / 𝑥⦌𝐵 = ∅) | ||
Theorem | csb0 4338 | The proper substitution of a class into the empty set is the empty set. (Contributed by NM, 18-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌∅ = ∅ | ||
Theorem | sbcel12 4339 | Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | sbceqg 4340 | Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | sbceqi 4341 | Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
⊢ 𝐴 ∈ V & ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷 & ⊢ ⦋𝐴 / 𝑥⦌𝐶 = 𝐸 ⇒ ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸) | ||
Theorem | sbcnel12g 4342 | Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | sbcne12 4343 | Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.) |
⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | sbcel1g 4344* | Move proper substitution in and out of a membership relation. Note that the scope of [𝐴 / 𝑥] is the wff 𝐵 ∈ 𝐶, whereas the scope of ⦋𝐴 / 𝑥⦌ is the class 𝐵. (Contributed by NM, 10-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝐶)) | ||
Theorem | sbceq1g 4345* | Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | ||
Theorem | sbcel2 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]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | sbceq2g 4347* | Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | csbcom 4348* | Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 | ||
Theorem | sbcnestgfw 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | ||
Theorem | csbnestgfw 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) | ||
Theorem | sbcnestgw 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.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | ||
Theorem | csbnestgw 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.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) | ||
Theorem | sbcco3gw 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 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) | ||
Theorem | sbcnestgf 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | ||
Theorem | csbnestgf 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) | ||
Theorem | sbcnestg 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.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | ||
Theorem | csbnestg 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.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶) | ||
Theorem | sbcco3g 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 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) | ||
Theorem | csbco3g 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 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷) | ||
Theorem | csbnest1g 4360 | Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶) | ||
Theorem | csbidm 4361* | Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) (Revised by NM, 18-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 | ||
Theorem | csbvarg 4362 | The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) | ||
Theorem | csbvargi 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 / 𝑥⦌𝑥 = 𝐴 | ||
Theorem | sbccsb 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]𝜑 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑}) | ||
Theorem | sbccsb2 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]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}) | ||
Theorem | rspcsbela 4366* | Special case related to rspsbc 3808. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷) | ||
Theorem | sbnfc2 4367* | Two ways of expressing "𝑥 is (effectively) not free in 𝐴". (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | ||
Theorem | csbab 4368* | Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 19-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑} | ||
Theorem | csbun 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⦌(𝐵 ∪ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | csbin 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⦌(𝐵 ∩ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | csbie2df 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.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐶) & ⊢ (𝜑 → Ⅎ𝑥𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷) | ||
Theorem | 2nreu 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ≠ 𝐵) → ((𝜓 ∧ 𝜒) → ¬ ∃!𝑥 ∈ 𝑋 𝜑)) | ||
Theorem | un00 4373 | Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.) |
⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅) | ||
Theorem | vss 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 ⊆ 𝐴 ↔ 𝐴 = V) | ||
Theorem | 0pss 4375 | The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.) |
⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅) | ||
Theorem | npss0 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 ⊊ ∅ | ||
Theorem | pssv 4377 | Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.) |
⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V) | ||
Theorem | disj 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.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | ||
Theorem | disjOLD 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.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | ||
Theorem | disjr 4380* | Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ 𝐴) | ||
Theorem | disj1 4381* | Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | ||
Theorem | reldisj 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.) |
⊢ (𝐴 ⊆ 𝐶 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵))) | ||
Theorem | reldisjOLD 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.) |
⊢ (𝐴 ⊆ 𝐶 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵))) | ||
Theorem | disj3 4384 | Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) | ||
Theorem | disjne 4385 | Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷) | ||
Theorem | disjeq0 4386 | Two disjoint sets are equal iff both are empty. (Contributed by AV, 19-Jun-2022.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅))) | ||
Theorem | disjel 4387 | A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.) |
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶 ∈ 𝐵) | ||
Theorem | disj2 4388 | Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)) | ||
Theorem | disj4 4389 | Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) | ||
Theorem | ssdisj 4390 | Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) | ||
Theorem | disjpss 4391 | A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.) |
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ 𝐵)) | ||
Theorem | undisj1 4392 | The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.) |
⊢ (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐶) = ∅) ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ∅) | ||
Theorem | undisj2 4393 | The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.) |
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ∩ 𝐶) = ∅) ↔ (𝐴 ∩ (𝐵 ∪ 𝐶)) = ∅) | ||
Theorem | ssindif0 4394 | Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅) | ||
Theorem | inelcm 4395 | The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) | ||
Theorem | minel 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.) |
⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶) | ||
Theorem | undif4 4397 | Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝐴 ∩ 𝐶) = ∅ → (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶)) | ||
Theorem | disjssun 4398 | Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶)) | ||
Theorem | vdif0 4399 | Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.) |
⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅) | ||
Theorem | difrab0eq 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 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅ ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑}) |
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