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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | abbi 2901 | 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 | abid1 2902* |
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 2767. The proof does not rely on cvjust 2767, so cvjust 2767
could be proved as a special instance of it. Note however that abid1 2902
necessarily relies on df-clel 2840, whereas cvjust 2767 does not.
This theorem requires ax-ext 2745, df-clab 2754, df-cleq 2765, df-clel 2840, 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 1506, cab 2753 and statements corresponding to defined class constructors. Note on the simultaneous presence in set.mm of this abid1 2902 and its commuted form abid2 2903: 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 2796 (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 11496 versus 1p1e2 11565. We do not need 1p1e2 11565, 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 11583, 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 computational form is often more natural. The situation is similar with biconditionals in propositional calculus: see for instance pm4.24 556 and anidm 557, 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 2902 and abid2 2903 are in set.mm. (Contributed by NM, 26-Dec-1993.) (Revised by BJ, 10-Nov-2020.) |
⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | ||
Theorem | abid2 2903* | A simplification of class abstraction. Commuted form of abid1 2902. See comments there. (Contributed by NM, 26-Dec-1993.) |
⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | ||
Theorem | cbvabv 2904* | Rule used to change bound variables, using implicit substitution. Version of cbvab 2905 with disjoint variable conditions requiring fewer axioms. (Contributed by NM, 26-May-1999.) Require 𝑥, 𝑦 be disjoint to avoid ax-11 2091 and ax-13 2299. (Revised by Steven Nguyen, 4-Dec-2022.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | ||
Theorem | cbvab 2905 | 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 | cbvabvOLD 2906* | Obsolete version of cbvabv 2904 as of 9-May-2023. (Contributed by NM, 26-May-1999.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | ||
Theorem | clelab 2907* | 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 2908* | 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 2909* | 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 2910 | Extend wff definition to include the not-free predicate for classes. |
wff Ⅎ𝑥𝐴 | ||
Theorem | nfcjust 2911* | Justification theorem for df-nfc 2912. (Contributed by Mario Carneiro, 13-Oct-2016.) |
⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) | ||
Definition | df-nfc 2912* | 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 1747 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfci 2913* | 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 2914* | 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 2915* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfcriv 2916* | Consequence of the not-free predicate, similiar to nfcri 2920. Requires 𝑦 and 𝐴 be disjoint, but is not based on ax-13 2299. (Contributed by Wolf Lammen, 13-May-2023.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | ||
Theorem | nfcd 2917* | 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 | nfcrd 2918* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfcrii 2919* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfcri 2920* | Consequence of the not-free predicate. (Note that unlike nfcr 2915, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | ||
Theorem | nfceqdf 2921 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) | ||
Theorem | nfceqi 2922 | Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) Avoid ax-12 2104. (Revised by Wolf Lammen, 19-Jun-2023.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) | ||
Theorem | nfceqiOLD 2923 | Obsolete proof of nfceqi 2922 as of 19-Jun-2023. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) | ||
Theorem | nfcxfr 2924 | 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 2925 | 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 2926* | If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 | ||
Theorem | nfcvd 2927* | If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) | ||
Theorem | nfab1 2928 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | ||
Theorem | nfnfc1 2929 | The setvar 𝑥 is bound in Ⅎ𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | ||
Theorem | clelsb3f 2930 | Substitution applied to an atomic wff (class version of elsb3 2406). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) (Proof shortened by Wolf Lammen, 7-May-2023.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | ||
Theorem | clelsb3fOLD 2931 | Obsolete version of clelsb3f 2930 as of 7-May-2023. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | ||
Theorem | nfab 2932 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑} | ||
Theorem | nfaba1 2933 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} | ||
Theorem | nfeqd 2934 | Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) | ||
Theorem | nfeld 2935 | Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) | ||
Theorem | nfnfc 2936 | Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2299. (Revised by Wolf Lammen, 10-Dec-2019.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥Ⅎ𝑦𝐴 | ||
Theorem | nfeq 2937 | 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 2938 | 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 2939* | Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel1 2940* | Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfeq2 2941* | Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel2 2942* | Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | drnfc1 2943 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-11 2091. (Revised by Wolf Lammen, 10-May-2023.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵)) | ||
Theorem | drnfc1OLD 2944 | Obsolete version of drnfc1 2943 as of 10-May-2023. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵)) | ||
Theorem | drnfc2 2945 | Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv 1879 with dral2 2372, leading to a one byte longer proof. However feel free to manually edit it according to conventions. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵)) | ||
Theorem | nfabd 2946 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2057 and ax-ext 2745. (Revised by Wolf Lammen, 23-May-2023.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | ||
Theorem | nfabd2 2947 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof shortened by Wolf Lammen, 10-May-2023.) |
⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | ||
Theorem | nfabd2OLD 2948 | Obsolete version of nfabd2 2947 as of 23-May-2023. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | ||
Theorem | nfabdOLD 2949 | Obsolete version of nfabd 2946 as of 10-May-2023. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | ||
Theorem | dvelimdc 2950 | Deduction form of dvelimc 2951. (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 2951 | Version of dvelim 2385 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑧𝐵 & ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵) | ||
Theorem | nfcvf 2952 | If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-ext 2745. (Revised by Wolf Lammen, 10-May-2023.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | ||
Theorem | nfcvf2 2953 | 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 | nfcvfOLD 2954 | Obsolete version of nfcvf 2952 as of 10-May-2023. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | ||
Theorem | cleqf 2955 | Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2883. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) Avoid ax-13 2299. (Revised by Wolf Lammen, 10-May-2023.) Avoid ax-10 2077. (Revised by Gino Giotto, 20-Aug-2023.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | cleqfOLD 2956 | Obsolete version of cleqf 2955 as of 10-May-2023. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | abid2f 2957 | 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 2958 | 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.) Avoid ax-13 2299. (Revised by Wolf Lammen, 13-May-2023.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | ||
Theorem | abeq2fOLD 2959 | 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.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | ||
Theorem | sbabel 2960* | 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 2961 | Extend wff notation to include inequality. |
wff 𝐴 ≠ 𝐵 | ||
Definition | df-ne 2962 | Define inequality. (Contributed by NM, 26-May-1993.) |
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | ||
Theorem | neii 2963 | Inference associated with df-ne 2962. (Contributed by BJ, 7-Jul-2018.) |
⊢ 𝐴 ≠ 𝐵 ⇒ ⊢ ¬ 𝐴 = 𝐵 | ||
Theorem | neir 2964 | Inference associated with df-ne 2962. (Contributed by BJ, 7-Jul-2018.) |
⊢ ¬ 𝐴 = 𝐵 ⇒ ⊢ 𝐴 ≠ 𝐵 | ||
Theorem | nne 2965 | Negation of inequality. (Contributed by NM, 9-Jun-2006.) |
⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | ||
Theorem | neneqd 2966 | Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | ||
Theorem | neneq 2967 | From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵) | ||
Theorem | neqned 2968 | If it is not the case that two classes are equal, then they are unequal. Converse of neneqd 2966. One-way deduction form of df-ne 2962. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2987. (Revised by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | neqne 2969 | From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | ||
Theorem | neirr 2970 | No class is unequal to itself. Inequality is irreflexive. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ ¬ 𝐴 ≠ 𝐴 | ||
Theorem | exmidne 2971 | Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) |
⊢ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵) | ||
Theorem | eqneqall 2972 | A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑)) | ||
Theorem | nonconne 2973 | Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.) |
⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) | ||
Theorem | necon3ad 2974 | Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓)) | ||
Theorem | necon3bd 2975 | Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓)) ⇒ ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵)) | ||
Theorem | necon2ad 2976 | Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) ⇒ ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) | ||
Theorem | necon2bd 2977 | Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.) |
⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) | ||
Theorem | necon1ad 2978 | Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (¬ 𝜓 → 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝜓)) | ||
Theorem | necon1bd 2979 | Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝜓)) ⇒ ⊢ (𝜑 → (¬ 𝜓 → 𝐴 = 𝐵)) | ||
Theorem | necon4ad 2980 | Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓)) ⇒ ⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵)) | ||
Theorem | necon4bd 2981 | Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓)) | ||
Theorem | necon3d 2982 | Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.) |
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 = 𝐷)) ⇒ ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵)) | ||
Theorem | necon1d 2983 | Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)) ⇒ ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)) | ||
Theorem | necon2d 2984 | Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.) |
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)) ⇒ ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)) | ||
Theorem | necon4d 2985 | Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)) ⇒ ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵)) | ||
Theorem | necon3ai 2986 | Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑) | ||
Theorem | necon3bi 2987 | Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝐴 = 𝐵 → 𝜑) ⇒ ⊢ (¬ 𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | necon1ai 2988 | Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (¬ 𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝐴 ≠ 𝐵 → 𝜑) | ||
Theorem | necon1bi 2989 | Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝐴 ≠ 𝐵 → 𝜑) ⇒ ⊢ (¬ 𝜑 → 𝐴 = 𝐵) | ||
Theorem | necon2ai 2990 | Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝐴 = 𝐵 → ¬ 𝜑) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | necon2bi 2991 | Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝐴 = 𝐵 → ¬ 𝜑) | ||
Theorem | necon4ai 2992 | Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | necon3i 2993 | Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.) (Proof shortened by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷) ⇒ ⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵) | ||
Theorem | necon1i 2994 | Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) |
⊢ (𝐴 ≠ 𝐵 → 𝐶 = 𝐷) ⇒ ⊢ (𝐶 ≠ 𝐷 → 𝐴 = 𝐵) | ||
Theorem | necon2i 2995 | Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) |
⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷) ⇒ ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵) | ||
Theorem | necon4i 2996 | Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
⊢ (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷) ⇒ ⊢ (𝐶 = 𝐷 → 𝐴 = 𝐵) | ||
Theorem | necon3abid 2997 | Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.) |
⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓)) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓)) | ||
Theorem | necon3bbid 2998 | Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.) |
⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵)) | ||
Theorem | necon1abid 2999 | Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓)) | ||
Theorem | necon1bbid 3000 | Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.) |
⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓)) ⇒ ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵)) |
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