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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ralrab 3701* | Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) | ||
Theorem | rexab 3702* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) Reduce axiom usage. (Revised by GG, 2-Nov-2024.) |
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒)) | ||
Theorem | rexabOLD 3703* | Obsolete version of rexab 3702 as of 2-Nov-2024. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒)) | ||
Theorem | rexrab 3704* | Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) | ||
Theorem | ralab2 3705* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2107. (Revised by GG, 1-Dec-2023.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒)) | ||
Theorem | ralrab2 3706* | Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒)) | ||
Theorem | rexab2 3707* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2107. (Revised by GG, 1-Dec-2023.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒)) | ||
Theorem | rexrab2 3708* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒)) | ||
Theorem | reurab 3709* | Restricted existential uniqueness of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒 ↔ ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒)) | ||
Theorem | abidnf 3710* | Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.) |
⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | ||
Theorem | dedhb 3711* | A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 2148 and nfab 2908 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 3710 is useful. (Contributed by NM, 8-Dec-2006.) |
⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓)) & ⊢ 𝜓 ⇒ ⊢ (Ⅎ𝑥𝐴 → 𝜑) | ||
Theorem | class2seteq 3712* | Writing a set as a class abstraction. This theorem looks artificial but was added to characterize the class abstraction whose existence is proved in class2set 5360. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) | ||
Theorem | nelrdva 3713* | Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) | ||
Theorem | eqeu 3714* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃!𝑥𝜑) | ||
Theorem | moeq 3715* | There exists at most one set equal to a given class. (Contributed by NM, 8-Mar-1995.) Shorten combined proofs of moeq 3715 and eueq 3716. (Proof shortened by BJ, 24-Sep-2022.) |
⊢ ∃*𝑥 𝑥 = 𝐴 | ||
Theorem | eueq 3716* | A class is a set if and only if there exists a unique set equal to it. (Contributed by NM, 25-Nov-1994.) Shorten combined proofs of moeq 3715 and eueq 3716. (Proof shortened by BJ, 24-Sep-2022.) |
⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) | ||
Theorem | eueqi 3717* | There exists a unique set equal to a given set. Inference associated with euequ 2594. See euequ 2594 in the case of a setvar. (Contributed by NM, 5-Apr-1995.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃!𝑥 𝑥 = 𝐴 | ||
Theorem | eueq2 3718* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)) | ||
Theorem | eueq3 3719* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ ¬ (𝜑 ∧ 𝜓) ⇒ ⊢ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) | ||
Theorem | moeq3 3720* | "At most one" property of equality (split into 3 cases). (The first two hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ ¬ (𝜑 ∧ 𝜓) ⇒ ⊢ ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) | ||
Theorem | mosub 3721* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
⊢ ∃*𝑥𝜑 ⇒ ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑) | ||
Theorem | mo2icl 3722* | Theorem for inferring "at most one". (Contributed by NM, 17-Oct-1996.) |
⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑) | ||
Theorem | mob2 3723* | Consequence of "at most one". (Contributed by NM, 2-Jan-2015.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓)) | ||
Theorem | moi2 3724* | Consequence of "at most one". (Contributed by NM, 29-Jun-2008.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝐴) | ||
Theorem | mob 3725* | Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒)) | ||
Theorem | moi 3726* | Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝐴 = 𝐵) | ||
Theorem | morex 3727* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) | ||
Theorem | euxfr2w 3728* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr2 3730 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2374. (Revised by GG, 10-Jan-2024.) |
⊢ 𝐴 ∈ V & ⊢ ∃*𝑦 𝑥 = 𝐴 ⇒ ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑) | ||
Theorem | euxfrw 3729* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr 3731 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2374. (Revised by GG, 10-Jan-2024.) |
⊢ 𝐴 ∈ V & ⊢ ∃!𝑦 𝑥 = 𝐴 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
Theorem | euxfr2 3730* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker euxfr2w 3728 when possible. (Contributed by NM, 14-Nov-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ ∃*𝑦 𝑥 = 𝐴 ⇒ ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑) | ||
Theorem | euxfr 3731* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker euxfrw 3729 when possible. (Contributed by NM, 14-Nov-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ ∃!𝑦 𝑥 = 𝐴 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
Theorem | euind 3732* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧∀𝑥(𝜑 → 𝑧 = 𝐴)) | ||
Theorem | reu2 3733* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | reu6 3734* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) | ||
Theorem | reu3 3735* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))) | ||
Theorem | reu6i 3736* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) | ||
Theorem | eqreu 3737* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐵 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmo4 3738* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | reu4 3739* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) | ||
Theorem | reu7 3740* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) | ||
Theorem | reu8 3741* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) | ||
Theorem | rmo3f 3742* | Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | rmo4f 3743* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | reu2eqd 3744* | Deduce equality from restricted uniqueness, deduction version. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜃)) & ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
Theorem | reueq 3745* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵) | ||
Theorem | rmoeq 3746* | Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by NM, 29-Oct-2020.) |
⊢ ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴 | ||
Theorem | rmoan 3747 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑)) | ||
Theorem | rmoim 3748 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | rmoimia 3749 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmoimi 3750 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmoimi2 3751 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐵 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | 2reu5a 3752 | Double restricted existential uniqueness in terms of restricted existence and restricted "at most one". (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑) ∧ ∃*𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑))) | ||
Theorem | reuimrmo 3753 | Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2613. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | 2reuswap 3754* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | 2reuswap2 3755* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | reuxfrd 3756* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 16-Jan-2012.) Separate variables 𝐵 and 𝐶. (Revised by Thierry Arnoux, 8-Oct-2017.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓)) | ||
Theorem | reuxfr 3757* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) ⇒ ⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑) | ||
Theorem | reuxfr1d 3758* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr1ds 3759. (Contributed by Thierry Arnoux, 7-Apr-2017.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | reuxfr1ds 3759* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 5424 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | reuxfr1 3760* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5425 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓) | ||
Theorem | reuind 3761* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ ((∀𝑥∀𝑦(((𝐴 ∈ 𝐶 ∧ 𝜑) ∧ (𝐵 ∈ 𝐶 ∧ 𝜓)) → 𝐴 = 𝐵) ∧ ∃𝑥(𝐴 ∈ 𝐶 ∧ 𝜑)) → ∃!𝑧 ∈ 𝐶 ∀𝑥((𝐴 ∈ 𝐶 ∧ 𝜑) → 𝑧 = 𝐴)) | ||
Theorem | 2rmorex 3762* | Double restricted quantification with "at most one", analogous to 2moex 2637. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | 2reu5lem1 3763* | Lemma for 2reu5 3766. Note that ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵𝜑 does not mean "there is exactly one 𝑥 in 𝐴 and exactly one 𝑦 in 𝐵 such that 𝜑 holds"; see comment for 2eu5 2653. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) | ||
Theorem | 2reu5lem2 3764* | Lemma for 2reu5 3766. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) | ||
Theorem | 2reu5lem3 3765* | Lemma for 2reu5 3766. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3895. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | ||
Theorem | 2reu5 3766* | Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2653 and reu3 3735. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | ||
Theorem | 2reurmo 3767 | Double restricted quantification with restricted existential uniqueness and restricted "at most one", analogous to 2eumo 2639. (Contributed by Alexander van der Vekens, 24-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | ||
Theorem | 2reurex 3768* | Double restricted quantification with existential uniqueness, analogous to 2euex 2638. (Contributed by Alexander van der Vekens, 24-Jun-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) | ||
Theorem | 2rmoswap 3769* | A condition allowing to swap restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2641. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | 2rexreu 3770* | Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2643. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | ||
This is a very useless definition, which "abbreviates" (𝑥 = 𝑦 → 𝜑) as CondEq(𝑥 = 𝑦 → 𝜑). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows to give a name to the specific ternary operation (𝑥 = 𝑦 → 𝜑). This is all used as part of a metatheorem: we want to say that ⊢ (𝑥 = 𝑦 → (𝜑(𝑥) ↔ 𝜑(𝑦))) and ⊢ (𝑥 = 𝑦 → 𝐴(𝑥) = 𝐴(𝑦)) are provable, for any expressions 𝜑(𝑥) or 𝐴(𝑥) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations. The metatheorem comes with a disjoint variables assumption: every variable in 𝜑(𝑥) is assumed disjoint from 𝑥 except 𝑥 itself. For such a proof by induction, we must consider each of the possible forms of 𝜑(𝑥). If it is a variable other than 𝑥, then we have CondEq(𝑥 = 𝑦 → 𝐴 = 𝐴) or CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜑)), which is provable by cdeqth 3775 and reflexivity. Since we are only working with class and wff expressions, it can't be 𝑥 itself in set.mm, but if it was we'd have to also prove CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) (where set equality is being used on the right). Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to 𝑥 or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one setvar variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the universal quantifier and the class builder). In each of the primitive proofs, we are allowed to assume that 𝑦 is disjoint from 𝜑(𝑥) and vice versa, because this is maintained through the induction. This is how we satisfy the disjoint variable conditions of cdeqab1 3780 and cdeqab 3778. | ||
Syntax | wcdeq 3771 | Extend wff notation to include conditional equality. This is a technical device used in the proof that Ⅎ is the not-free predicate, and that definitions are conservative as a result. |
wff CondEq(𝑥 = 𝑦 → 𝜑) | ||
Definition | df-cdeq 3772 | Define conditional equality. All the notation to the left of the ↔ is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq𝑥𝑦𝜑. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) | ||
Theorem | cdeqi 3773 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqri 3774 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqth 3775 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ 𝜑 ⇒ ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | ||
Theorem | cdeqnot 3776 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) | ||
Theorem | cdeqal 3777* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓)) | ||
Theorem | cdeqab 3778* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → {𝑧 ∣ 𝜑} = {𝑧 ∣ 𝜓}) | ||
Theorem | cdeqal1 3779* | Distribute conditional equality over quantification. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
Theorem | cdeqab1 3780* | Distribute conditional equality over abstraction. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}) | ||
Theorem | cdeqim 3781 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ CondEq(𝑥 = 𝑦 → (𝜒 ↔ 𝜃)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) | ||
Theorem | cdeqcv 3782 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) | ||
Theorem | cdeqeq 3783 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | cdeqel 3784 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | ||
Theorem | nfcdeq 3785* | If we have a conditional equality proof, where 𝜑 is 𝜑(𝑥) and 𝜓 is 𝜑(𝑦), and 𝜑(𝑥) in fact does not have 𝑥 free in it according to Ⅎ, then 𝜑(𝑥) ↔ 𝜑(𝑦) unconditionally. This proves that Ⅎ𝑥𝜑 is actually a not-free predicate. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜑 ↔ 𝜓) | ||
Theorem | nfccdeq 3786* | Variation of nfcdeq 3785 for classes. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2154. (Revised by GG, 19-May-2023.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ 𝐴 = 𝐵 | ||
Theorem | rru 3787* |
Relative version of Russell's paradox ru 3788 (which corresponds to the
case 𝐴 = V).
Originally a subproof in pwnss 5357. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3044. (Revised by Steven Nguyen, 23-Nov-2022.) Reduce axiom usage. (Revised by GG, 30-Aug-2024.) |
⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 | ||
Theorem | ru 3788 |
Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.
In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as 𝐴 ∈ V, asserted that any collection of sets 𝐴 is a set i.e. belongs to the universe V of all sets. In particular, by substituting {𝑥 ∣ 𝑥 ∉ 𝑥} (the "Russell class") for 𝐴, it asserted {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system, which Frege acknowledged in the second edition of his Grundgesetze der Arithmetik. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 5326 asserting that 𝐴 is a set only when it is smaller than some other set 𝐵. However, Zermelo was then faced with a "chicken and egg" problem of how to show 𝐵 is a set, leading him to introduce the set-building axioms of Null Set 0ex 5312, Pairing prex 5442, Union uniex 7759, Power Set pwex 5385, and Infinity omex 9680 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 6655 (whose modern formalization is due to Skolem, also in 1922). Thus, in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics! Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than setvar variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287). Russell himself continued in a different direction, avoiding the paradox with his "theory of types". Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contrarily to ZF and NBG set theories. Russell's paradox has other consequences: when classes are too large (beyond the size of those used in standard mathematics), the axiom of choice ac4 10512 and Cantor's theorem canth 7384 are provably false. (See ncanth 7385 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 5301 replaces ax-rep 5284) with ax-sep 5301 restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic", J. Symb. Logic 9:1-19 (1944). Under our ZF set theory, every set is a member of the Russell class by elirrv 9633 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9639). See ruALT 9640 for an alternate proof of ru 3788 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2374. (Revised by BJ, 12-Oct-2019.) Remove use of ax-10 2138, ax-11 2154, and ax-12 2174. (Revised by BTernaryTau, 20-Jun-2025.) (Proof modification is discouraged.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Theorem | ruOLD 3789 | Obsolete version of ru 3788 as of 20-Jun-2025. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2374. (Revised by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Syntax | wsbc 3790 | Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class 𝐴 for setvar variable 𝑥 in wff 𝜑". |
wff [𝐴 / 𝑥]𝜑 | ||
Definition | df-sbc 3791 |
Define the proper substitution of a class for a set.
When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3800). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3822 for our definition, whose right-hand side always evaluates to true for proper classes. Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 3792 below). For example, if 𝐴 is a proper class, Quine's substitution of 𝐴 for 𝑦 in 0 ∈ 𝑦 evaluates to 0 ∈ 𝐴 rather than our falsehood. (This can be seen by substituting 𝐴, 𝑦, and 0 for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactic breakdown of 𝜑, and it does not seem possible to express it with a single closed formula. If we did not want to commit to any specific proper class behavior, we could use this definition only to prove Theorem dfsbcq 3792, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3791 in the form of sbc8g 3798. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of 𝐴 in every use of this definition) we allow direct reference to df-sbc 3791 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. Theorem sbc2or 3799 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3792. The related definition df-csb 3908 defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | dfsbcq 3792 |
Proper substitution of a class for a set in a wff given equal classes.
This is the essence of the sixth axiom of Frege, specifically Proposition
52 of [Frege1879] p. 50.
This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 3791 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 3793 instead of df-sbc 3791. (dfsbcq2 3793 is needed because unlike Quine we do not overload the df-sb 2062 syntax.) As a consequence of these theorems, we can derive sbc8g 3798, which is a weaker version of df-sbc 3791 that leaves substitution undefined when 𝐴 is a proper class. However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3798, so we will allow direct use of df-sbc 3791 after Theorem sbc2or 3799 below. Proper substitution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.) |
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑)) | ||
Theorem | dfsbcq2 3793 | This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 2062 and substitution for class variables df-sbc 3791. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3792. (Contributed by NM, 31-Dec-2016.) |
⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbsbc 3794 | Show that df-sb 2062 and df-sbc 3791 are equivalent when the class term 𝐴 in df-sbc 3791 is a setvar variable. This theorem lets us reuse theorems based on df-sb 2062 for proofs involving df-sbc 3791. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | sbceq1d 3795 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) | ||
Theorem | sbceq1dd 3796 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) ⇒ ⊢ (𝜑 → [𝐵 / 𝑥]𝜓) | ||
Theorem | sbceqbid 3797* | Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) | ||
Theorem | sbc8g 3798 | This is the closest we can get to df-sbc 3791 if we start from dfsbcq 3792 (see its comments) and dfsbcq2 3793. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | ||
Theorem | sbc2or 3799* | The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for [𝐴 / 𝑥]𝜑 behavior at proper classes, matching the sbc5 3818 (false) and sbc6 3822 (true) conclusions. This is interesting since dfsbcq 3792 and dfsbcq2 3793 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem does not tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable 𝑦 that 𝜑 or 𝐴 may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.) |
⊢ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | ||
Theorem | sbcex 3800 | By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
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