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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | rexrab 3701* | 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 3702* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2110. (Revised by GG, 1-Dec-2023.) |
| ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒)) | ||
| Theorem | ralrab2 3703* | 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 3704* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2110. (Revised by GG, 1-Dec-2023.) |
| ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒)) | ||
| Theorem | rexrab2 3705* | 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 3706* | 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 3707* | 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 3708* | A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 2151 and nfab 2910 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 3707 is useful. (Contributed by NM, 8-Dec-2006.) |
| ⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓)) & ⊢ 𝜓 ⇒ ⊢ (Ⅎ𝑥𝐴 → 𝜑) | ||
| Theorem | class2seteq 3709* | 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 5353. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.) |
| ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) | ||
| Theorem | nelrdva 3710* | Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) | ||
| Theorem | eqeu 3711* | 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 3712* | There exists at most one set equal to a given class. (Contributed by NM, 8-Mar-1995.) Shorten combined proofs of moeq 3712 and eueq 3713. (Proof shortened by BJ, 24-Sep-2022.) |
| ⊢ ∃*𝑥 𝑥 = 𝐴 | ||
| Theorem | eueq 3713* | 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 3712 and eueq 3713. (Proof shortened by BJ, 24-Sep-2022.) |
| ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) | ||
| Theorem | eueqi 3714* | There exists a unique set equal to a given set. Inference associated with euequ 2596. See euequ 2596 in the case of a setvar. (Contributed by NM, 5-Apr-1995.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ∃!𝑥 𝑥 = 𝐴 | ||
| Theorem | eueq2 3715* | 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 3716* | 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 3717* | "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 3718* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
| ⊢ ∃*𝑥𝜑 ⇒ ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑) | ||
| Theorem | mo2icl 3719* | Theorem for inferring "at most one". (Contributed by NM, 17-Oct-1996.) |
| ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑) | ||
| Theorem | mob2 3720* | Consequence of "at most one". (Contributed by NM, 2-Jan-2015.) |
| ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓)) | ||
| Theorem | moi2 3721* | Consequence of "at most one". (Contributed by NM, 29-Jun-2008.) |
| ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝐴) | ||
| Theorem | mob 3722* | 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 3723* | 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 3724* | 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 3725* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr2 3727 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2376. (Revised by GG, 10-Jan-2024.) |
| ⊢ 𝐴 ∈ V & ⊢ ∃*𝑦 𝑥 = 𝐴 ⇒ ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑) | ||
| Theorem | euxfrw 3726* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr 3728 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2376. (Revised by GG, 10-Jan-2024.) |
| ⊢ 𝐴 ∈ V & ⊢ ∃!𝑦 𝑥 = 𝐴 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
| Theorem | euxfr2 3727* | 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 2376. Use the weaker euxfr2w 3725 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 3728* | 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 2376. Use the weaker euxfrw 3726 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 3729* | 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 3730* | 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 3731* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
| ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) | ||
| Theorem | reu3 3732* | 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 3733* | 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 3734* | 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 3735* | 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 3736* | 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 3737* | 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 3738* | 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 3739* | 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 3740* | 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 3741* | 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 3742* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
| ⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵) | ||
| Theorem | rmoeq 3743* | 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 3744 | 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 3745 | 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 3746 | 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 3747 | 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 3748 | 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 3749 | 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 3750 | Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2615. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
| ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) | ||
| Theorem | 2reuswap 3751* | 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 3752* | 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 3753* | 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 3754* | 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 3755* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr1ds 3756. (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 3756* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 5417 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 3757* | Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5418 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 3758* | 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 3759* | Double restricted quantification with "at most one", analogous to 2moex 2639. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
| ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑) | ||
| Theorem | 2reu5lem1 3760* | Lemma for 2reu5 3763. 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 2655. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
| ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) | ||
| Theorem | 2reu5lem2 3761* | Lemma for 2reu5 3763. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
| ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑)) | ||
| Theorem | 2reu5lem3 3762* | Lemma for 2reu5 3763. 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 3886. (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 3763* | Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2655 and reu3 3732. (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 3764 | Double restricted quantification with restricted existential uniqueness and restricted "at most one", analogous to 2eumo 2641. (Contributed by Alexander van der Vekens, 24-Jun-2017.) |
| ⊢ (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | ||
| Theorem | 2reurex 3765* | Double restricted quantification with existential uniqueness, analogous to 2euex 2640. (Contributed by Alexander van der Vekens, 24-Jun-2017.) |
| ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) | ||
| Theorem | 2rmoswap 3766* | A condition allowing to swap restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2643. (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 3767* | Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2645. (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 3772 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 3777 and cdeqab 3775. | ||
| Syntax | wcdeq 3768 | 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 3769 | 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 3770 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
| ⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | ||
| Theorem | cdeqri 3771 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
| ⊢ CondEq(𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (𝑥 = 𝑦 → 𝜑) | ||
| Theorem | cdeqth 3772 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
| ⊢ 𝜑 ⇒ ⊢ CondEq(𝑥 = 𝑦 → 𝜑) | ||
| Theorem | cdeqnot 3773 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
| ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) | ||
| Theorem | cdeqal 3774* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
| ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓)) | ||
| Theorem | cdeqab 3775* | 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 3776* | Distribute conditional equality over quantification. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
| ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
| Theorem | cdeqab1 3777* | Distribute conditional equality over abstraction. Usage of this theorem is discouraged because it depends on ax-13 2376. (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 3778 | 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 3779 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
| ⊢ CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) | ||
| Theorem | cdeqeq 3780 | 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 3781 | 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 3782* | 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 2376. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜑 ↔ 𝜓) | ||
| Theorem | nfccdeq 3783* | Variation of nfcdeq 3782 for classes. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2157. (Revised by GG, 19-May-2023.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑥𝐴 & ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ 𝐴 = 𝐵 | ||
| Theorem | rru 3784* |
Relative version of Russell's paradox ru 3785 (which corresponds to the
case 𝐴 = V).
Originally a subproof in pwnss 5350. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3046. (Revised by Steven Nguyen, 23-Nov-2022.) Reduce axiom usage. (Revised by GG, 30-Aug-2024.) |
| ⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 | ||
| Theorem | ru 3785 |
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 5319 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 5305, Pairing prex 5435, Union uniex 7757, Power Set pwex 5378, and Infinity omex 9679 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 6653 (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 10511 and Cantor's theorem canth 7383 are provably false. (See ncanth 7384 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 5294 replaces ax-rep 5277) with ax-sep 5294 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 9632 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9638). See ruALT 9639 for an alternate proof of ru 3785 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2376. (Revised by BJ, 12-Oct-2019.) Remove use of ax-10 2141, ax-11 2157, and ax-12 2177. (Revised by BTernaryTau, 20-Jun-2025.) (Proof modification is discouraged.) |
| ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
| Theorem | ruOLD 3786 | Obsolete version of ru 3785 as of 20-Jun-2025. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2376. (Revised by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
| Syntax | wsbc 3787 | 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 3788 |
Define the proper substitution of a class for a set.
When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3797). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3818 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 3789 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 3789, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3788 in the form of sbc8g 3795. 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 3788 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. Theorem sbc2or 3796 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3789. The related definition df-csb 3899 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 3789 |
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 3788 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 3790 instead of df-sbc 3788. (dfsbcq2 3790 is needed because unlike Quine we do not overload the df-sb 2065 syntax.) As a consequence of these theorems, we can derive sbc8g 3795, which is a weaker version of df-sbc 3788 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 3795, so we will allow direct use of df-sbc 3788 after Theorem sbc2or 3796 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 3790 | 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 2065 and substitution for class variables df-sbc 3788. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3789. (Contributed by NM, 31-Dec-2016.) |
| ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
| Theorem | sbsbc 3791 | Show that df-sb 2065 and df-sbc 3788 are equivalent when the class term 𝐴 in df-sbc 3788 is a setvar variable. This theorem lets us reuse theorems based on df-sb 2065 for proofs involving df-sbc 3788. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
| ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
| Theorem | sbceq1d 3792 | 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 3793 | 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 3794* | 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 3795 | This is the closest we can get to df-sbc 3788 if we start from dfsbcq 3789 (see its comments) and dfsbcq2 3790. (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 3796* | 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 3815 (false) and sbc6 3818 (true) conclusions. This is interesting since dfsbcq 3789 and dfsbcq2 3790 (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 3797 | 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) | ||
| Theorem | sbceq1a 3798 | Equality theorem for class substitution. Class version of sbequ12 2251. (Contributed by NM, 26-Sep-2003.) |
| ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
| Theorem | sbceq2a 3799 | Equality theorem for class substitution. Class version of sbequ12r 2252. (Contributed by NM, 4-Jan-2017.) |
| ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
| Theorem | spsbc 3800 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. This is Frege's ninth axiom per Proposition 58 of [Frege1879] p. 51. See also stdpc4 2068 and rspsbc 3878. (Contributed by NM, 16-Jan-2004.) |
| ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | ||
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