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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | eqeltrd 2701 | Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eqeltrrd 2702 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝐶) | ||
Theorem | eleqtrd 2703 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eleqtrrd 2704 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl5eqel 2705 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl5eqelr 2706 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl5eleq 2707 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl5eleqr 2708 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl6eqel 2709 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl6eqelr 2710 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl6eleq 2711 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl6eleqr 2712 | A membership and equality inference. (Contributed by NM, 24-Apr-2005.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | 3eltr3i 2713 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐶 ∈ 𝐷 | ||
Theorem | 3eltr4i 2714 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ 𝐶 ∈ 𝐷 | ||
Theorem | 3eltr3d 2715 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) | ||
Theorem | 3eltr4d 2716 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐴) & ⊢ (𝜑 → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) | ||
Theorem | 3eltr3g 2717 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) | ||
Theorem | 3eltr4g 2718 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) | ||
Theorem | eleq2s 2719 | Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝐴 ∈ 𝐵 → 𝜑) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝐴 ∈ 𝐶 → 𝜑) | ||
Theorem | eqneltrd 2720 | If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) | ||
Theorem | eqneltrrd 2721 | If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) | ||
Theorem | neleqtrd 2722 | If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) | ||
Theorem | neleqtrrd 2723 | If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) |
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) | ||
Theorem | cleqh 2724* | Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2790. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.) Remove dependency on ax-13 2246. (Revised by BJ, 30-Nov-2020.) |
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) & ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | nelneq 2725 | A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.) |
⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵) | ||
Theorem | nelneq2 2726 | A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶) | ||
Theorem | eqsb3lem 2727* | Lemma for eqsb3 2728. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴) | ||
Theorem | eqsb3 2728* | Substitution applied to an atomic wff (class version of equsb3 2432). (Contributed by Rodolfo Medina, 28-Apr-2010.) |
⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴) | ||
Theorem | clelsb3 2729* | Substitution applied to an atomic wff (class version of elsb3 2434). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | ||
Theorem | hbxfreq 2730 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1752 for equivalence version. (Contributed by NM, 21-Aug-2007.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) ⇒ ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | ||
Theorem | hblem 2731* | Change the free variable of a hypothesis builder. Lemma for nfcrii 2757. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) |
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) ⇒ ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) | ||
Theorem | abeq2 2732* |
Equality of a class variable and a class abstraction (also called a
class builder). Theorem 5.1 of [Quine]
p. 34. This theorem shows the
relationship between expressions with class abstractions and expressions
with class variables. Note that abbi 2737 and its relatives are among
those useful for converting theorems with class variables to equivalent
theorems with wff variables, by first substituting a class abstraction
for each class variable.
Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable 𝜑 (that has a free variable 𝑥) to a theorem with a class variable 𝐴, we substitute 𝑥 ∈ 𝐴 for 𝜑 throughout and simplify, where 𝐴 is a new class variable not already in the wff. An example is the conversion of zfauscl 4783 to inex1 4799 (look at the instance of zfauscl 4783 that occurs in the proof of inex1 4799). Conversely, to convert a theorem with a class variable 𝐴 to one with 𝜑, we substitute {𝑥 ∣ 𝜑} for 𝐴 throughout and simplify, where 𝑥 and 𝜑 are new setvar and wff variables not already in the wff. Examples include dfsymdif2 3851 and cp 8754; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 8753. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 26-May-1993.) |
⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | ||
Theorem | abeq1 2733* | Equality of a class variable and a class abstraction. Commuted form of abeq2 2732. (Contributed by NM, 20-Aug-1993.) |
⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) | ||
Theorem | abeq2d 2734 | Equality of a class variable and a class abstraction (deduction form of abeq2 2732). (Contributed by NM, 16-Nov-1995.) |
⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) | ||
Theorem | abeq2i 2735 | Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.) (Proof shortened by Wolf Lammen, 15-Nov-2019.) |
⊢ 𝐴 = {𝑥 ∣ 𝜑} ⇒ ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) | ||
Theorem | abeq1i 2736 | Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Nov-2019.) |
⊢ {𝑥 ∣ 𝜑} = 𝐴 ⇒ ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴) | ||
Theorem | abbi 2737 | Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) | ||
Theorem | abbi2i 2738* | Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 26-May-1993.) |
⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) ⇒ ⊢ 𝐴 = {𝑥 ∣ 𝜑} | ||
Theorem | abbii 2739 | Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 26-May-1993.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} | ||
Theorem | abbid 2740 | Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | ||
Theorem | abbidv 2741* | Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | ||
Theorem | abbi2dv 2742* | Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) ⇒ ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) | ||
Theorem | abbi1dv 2743* | Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴) | ||
Theorem | abid1 2744* |
Every class is equal to a class abstraction (the class of sets belonging
to it). Theorem 5.2 of [Quine] p. 35.
This is a generalization to
classes of cvjust 2617. The proof does not rely on cvjust 2617, so cvjust 2617
could be proved as a special instance of it. Note however that abid1 2744
necessarily relies on df-clel 2618, whereas cvjust 2617 does not.
This theorem requires ax-ext 2602, df-clab 2609, df-cleq 2615, df-clel 2618, but to prove that any specific class term not containing class variables is a setvar or can be written as (is equal to) a class abstraction does not require these $a-statements. This last fact is a metatheorem, consequence of the fact that the only $a-statements with typecode class are cv 1482, cab 2608 and statements corresponding to defined class constructors. Note on the simultaneous presence in set.mm of this abid1 2744 and its commuted form abid2 2745: It is rare that two forms so closely related both appear in set.mm. Indeed, such equalities are generally used in later proofs as parts of transitive inferences, and with the many variants of eqtri 2644 (search for *eqtr*), it would be rare that either one would shorten a proof compared to the other. There is typically a choice between (what we call) a "definitional form" where the shorter expression is on the lhs, and a "computational form" where the shorter expression is on the rhs. An example is df-2 11079 versus 1p1e2 11134. We do not need 1p1e2 11134, but because it occurs "naturally" in computations, it can be useful to have it directly, together with a uniform set of 1-digit operations like 1p2e3 11152, etc. In most cases, we do not need both a definitional and a computational forms. A definitional form would favor consistency with genuine definitions, while a computationa form is often more natural. The situation is similar with biconditionals in propositional calculus: see for instance pm4.24 675 and anidm 676, while other biconditionals generally appear in a single form (either definitional, but more often computational). In the present case, the equality is important enough that both abid1 2744 and abid2 2745 are in set.mm. (Contributed by NM, 26-Dec-1993.) (Revised by BJ, 10-Nov-2020.) |
⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | ||
Theorem | abid2 2745* | A simplification of class abstraction. Commuted form of abid1 2744. See comments there. (Contributed by NM, 26-Dec-1993.) |
⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | ||
Theorem | cbvab 2746 | Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | ||
Theorem | cbvabv 2747* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | ||
Theorem | clelab 2748* | Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | ||
Theorem | clabel 2749* | Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.) |
⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))) | ||
Theorem | sbab 2750* | The right-hand side of the second equality is a way of representing proper substitution of 𝑦 for 𝑥 into a class variable. (Contributed by NM, 14-Sep-2003.) |
⊢ (𝑥 = 𝑦 → 𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}) | ||
Syntax | wnfc 2751 | Extend wff definition to include the not-free predicate for classes. |
wff Ⅎ𝑥𝐴 | ||
Theorem | nfcjust 2752* | Justification theorem for df-nfc 2753. (Contributed by Mario Carneiro, 13-Oct-2016.) |
⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) | ||
Definition | df-nfc 2753* | Define the not-free predicate for classes. This is read "𝑥 is not free in 𝐴". Not-free means that the value of 𝑥 cannot affect the value of 𝐴, e.g., any occurrence of 𝑥 in 𝐴 is effectively bound by a "for all" or something that expands to one (such as "there exists"). It is defined in terms of the not-free predicate df-nf 1710 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfci 2754* | Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 ⇒ ⊢ Ⅎ𝑥𝐴 | ||
Theorem | nfcii 2755* | Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) ⇒ ⊢ Ⅎ𝑥𝐴 | ||
Theorem | nfcr 2756* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfcrii 2757* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfcri 2758* | Consequence of the not-free predicate. (Note that unlike nfcr 2756, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | ||
Theorem | nfcd 2759* | Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝐴) | ||
Theorem | nfceqdf 2760 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) | ||
Theorem | nfceqi 2761 | Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) | ||
Theorem | nfcxfr 2762 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ 𝐴 = 𝐵 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥𝐴 | ||
Theorem | nfcxfrd 2763 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝐴) | ||
Theorem | nfcv 2764* | If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 | ||
Theorem | nfcvd 2765* | If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) | ||
Theorem | nfab1 2766 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | ||
Theorem | nfnfc1 2767 | The setvar 𝑥 is bound in Ⅎ𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | ||
Theorem | clelsb3f 2768 | Substitution applied to an atomic wff (class version of elsb3 2434). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | ||
Theorem | nfab 2769 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑} | ||
Theorem | nfaba1 2770 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} | ||
Theorem | nfcrd 2771* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfeqd 2772 | Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) | ||
Theorem | nfeld 2773 | Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) | ||
Theorem | nfnfc 2774 | Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2246. (Revised by Wolf Lammen, 10-Dec-2019.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥Ⅎ𝑦𝐴 | ||
Theorem | nfnfcALT 2775 | Alternate proof of nfnfc 2774. Shorter but requiring more axioms. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥Ⅎ𝑦𝐴 | ||
Theorem | nfeq 2776 | Hypothesis builder for equality. (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel 2777 | Hypothesis builder for elementhood. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfeq1 2778* | Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel1 2779* | Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfeq2 2780* | Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel2 2781* | Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | drnfc1 2782 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵)) | ||
Theorem | drnfc2 2783 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵)) | ||
Theorem | nfabd2 2784 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | ||
Theorem | nfabd 2785 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | ||
Theorem | dvelimdc 2786 | Deduction form of dvelimc 2787. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑧𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑧𝐵) & ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) | ||
Theorem | dvelimc 2787 | Version of dvelim 2337 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑧𝐵 & ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵) | ||
Theorem | nfcvf 2788 | If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | ||
Theorem | nfcvf2 2789 | If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) | ||
Theorem | cleqf 2790 | Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2724. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | abid2f 2791 | A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | ||
Theorem | abeq2f 2792 | Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a non-free variable in 𝐴 is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | ||
Theorem | sbabel 2793* | Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 26-Dec-2019.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴) | ||
Syntax | wne 2794 | Extend wff notation to include inequality. |
wff 𝐴 ≠ 𝐵 | ||
Definition | df-ne 2795 | Define inequality. (Contributed by NM, 26-May-1993.) |
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | ||
Theorem | neii 2796 | Inference associated with df-ne 2795. (Contributed by BJ, 7-Jul-2018.) |
⊢ 𝐴 ≠ 𝐵 ⇒ ⊢ ¬ 𝐴 = 𝐵 | ||
Theorem | neir 2797 | Inference associated with df-ne 2795. (Contributed by BJ, 7-Jul-2018.) |
⊢ ¬ 𝐴 = 𝐵 ⇒ ⊢ 𝐴 ≠ 𝐵 | ||
Theorem | nne 2798 | Negation of inequality. (Contributed by NM, 9-Jun-2006.) |
⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | ||
Theorem | neneqd 2799 | Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | ||
Theorem | neneq 2800 | From inequality to non equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵) |
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