Home Metamath Proof ExplorerTheorem List (p. 29 of 451) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28679) Hilbert Space Explorer (28680-30202) Users' Mathboxes (30203-45093)

Theorem List for Metamath Proof Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeleq1ab 2801 Extension (in the sense of Remark 3 of the comment of df-clab 2800) of elequ1 2122 from formulas of the form "setvar setvar" to formulas of the form "setvar class abstraction". This extension does not require ax-8 2117 contrary to elequ1 2122, but recall from Remark 3 of the comment of df-clab 2800 that it can be considered an extension only because of cvjust 2816, which does require ax-8 2117.

This is an instance of eleq1w 2894 where the containing class is a class abstraction, and contrary to it, it can be proved without df-clel 2892. See also eleq1 2899 for general classes.

The straightforward yet important fact that this statement can be proved from FOL= plus df-clab 2800 (hence without ax-ext 2793, df-cleq 2814 or df-clel 2892) was stressed by Mario Carneiro. (Contributed by BJ, 17-Aug-2023.)

(𝑥 = 𝑦 → (𝑥 ∈ {𝑧𝜑} ↔ 𝑦 ∈ {𝑧𝜑}))

Theoremcleljustab 2802* Extension of cleljust 2124 from formulas of the form "setvar setvar" to formulas of the form "setvar class abstraction". This is an instance of dfclel 2893 where the containing class is a class abstraction. The same remarks as for eleq1ab 2801 apply. (Contributed by BJ, 8-Nov-2021.) (Proof shortened by Steven Nguyen, 19-May-2023.)
(𝑥 ∈ {𝑦𝜑} ↔ ∃𝑧(𝑧 = 𝑥𝑧 ∈ {𝑦𝜑}))

Theoremabid 2803 Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 26-May-1993.)
(𝑥 ∈ {𝑥𝜑} ↔ 𝜑)

Theoremvexwt 2804 A standard theorem of predicate calculus (stdpc4 2074) expressed using class abstractions. Closed form of vexw 2805. (Contributed by BJ, 14-Jun-2019.)
(∀𝑥𝜑𝑦 ∈ {𝑥𝜑})

Theoremvexw 2805 If 𝜑 is a theorem, then any set belongs to the class {𝑥𝜑}. Therefore, {𝑥𝜑} is "a" universal class.

This is the closest one can get to defining a universal class, or proving vex 3474, without using ax-ext 2793. Note that this theorem has no disjoint variable condition and does not use df-clel 2892 nor df-cleq 2814 either: only propositional logic and ax-gen 1797 and df-clab 2800. This is sbt 2072 expressed using class abstractions.

Without ax-ext 2793, one cannot define "the" universal class, since one could not prove for instance the justification theorem {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} (see vjust 3472). Indeed, in order to prove any equality of classes, one needs df-cleq 2814, which has ax-ext 2793 as a hypothesis. Therefore, the classes {𝑥 ∣ ⊤}, {𝑦 ∣ (𝜑𝜑)}, {𝑧 ∣ (∀𝑡𝑡 = 𝑡 → ∀𝑡𝑡 = 𝑡)} and countless others are all universal classes whose equality cannot be proved without ax-ext 2793. Once dfcleq 2815 is available, we will define "the" universal class in df-v 3473.

Its degenerate instance is also a simple consequence of abid 2803 (using mpbir 234). (Contributed by BJ, 13-Jun-2019.) Reduce axiom dependencies. (Revised by Steven Nguyen, 25-Apr-2023.)

𝜑       𝑦 ∈ {𝑥𝜑}

Theoremvextru 2806 Every setvar is a member of {𝑥 ∣ ⊤}, which is therefore "a" universal class. Once class extensionality dfcleq 2815 is available, we can say "the" universal class (see df-v 3473). This is sbtru 2073 expressed using class abstractions. (Contributed by BJ, 2-Sep-2023.)
𝑦 ∈ {𝑥 ∣ ⊤}

Theoremhbab1 2807* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.)
(𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})

Theoremnfsab1 2808* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2178. (Revised by SN, 20-Sep-2023.)
𝑥 𝑦 ∈ {𝑥𝜑}

Theoremnfsab1OLD 2809* Obsolete version of nfsab1 2808 as of 20-Sep-2023. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 𝑦 ∈ {𝑥𝜑}

Theoremhbab 2810* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) Add disjoint variable condition to avoid ax-13 2391. See hbabg 2811 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝜑 → ∀𝑥𝜑)       (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})

Theoremhbabg 2811* Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2391. See hbab 2810 for a version with more disjoint variable conditions, but not requiring ax-13 2391. (Contributed by NM, 1-Mar-1995.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})

Theoremnfsab 2812* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2391. See nfsabg 2813 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝜑       𝑥 𝑧 ∈ {𝑦𝜑}

Theoremnfsabg 2813* Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2391. See nfsab 2812 for a version with more disjoint variable conditions, but not requiring ax-13 2391. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
𝑥𝜑       𝑥 𝑧 ∈ {𝑦𝜑}

2.1.2.2  Class equality

This section introduces class equality in df-cleq 2814.

Note that apart from the local introduction of class variables to state the syntax axioms wceq 1538 and wcel 2115, this section is the first to use class variables. Therefore, the file set.mm contains declarations of class variables at the beginning of this section (not visible on the webpages).

Definitiondf-cleq 2814* Define the equality connective between classes. Definition 2.7 of [Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4 provides its justification and methods for eliminating it. Note that its elimination will not necessarily result in a single wff in the original language but possibly a "scheme" of wffs.

The hypotheses express that all instances of the conclusion where class variables are replaced with setvar variables hold. Therefore, this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This is only a proof sketch of conservativity; for details see Appendix of [Levy] p. 357. This is the reason why we call this axiomatic statement a "definition", even though it does not have the usual form of a definition. If we required a definition to have the usual form, we would call df-cleq 2814 an axiom.

See also comments under df-clab 2800, df-clel 2892, and abeq2 2944.

In the form of dfcleq 2815, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms.

While the three class definitions df-clab 2800, df-cleq 2814, and df-clel 2892 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker.

For a general discussion of the theory of classes, see mmset.html#class 2892. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.)

(𝑦 = 𝑧 ↔ ∀𝑢(𝑢𝑦𝑢𝑧))    &   (𝑡 = 𝑡 ↔ ∀𝑣(𝑣𝑡𝑣𝑡))       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremdfcleq 2815* The defining characterization of class equality. It is proved, over Tarski's FOL, from the axiom of (set) extensionality (ax-ext 2793) and the definition of class equality (df-cleq 2814). Its forward implication is called "class extensionality". Remark: the proof uses axextb 2796 to prove also the hypothesis of df-cleq 2814 that is a degenerate instance, but it could be proved also from minimal propositional calculus and { ax-gen 1797, equid 2020 }. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.)
(𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremcvjust 2816* Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1537, which allows us to substitute a setvar variable for a class variable. See also cab 2799 and df-clab 2800. Note that this is not a rigorous justification, because cv 1537 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." See abid1 2953 for the version of cvjust 2816 extended to classes. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2391. (Revised by Wolf Lammen, 4-May-2023.)
𝑥 = {𝑦𝑦𝑥}

Theoremax9ALT 2817 Proof of ax-9 2125 from Tarski's FOL and dfcleq 2815. For a version not using ax-8 2117 either, see bj-ax9 34232. This shows that dfcleq 2815 is too powerful to be used as a definition instead of df-cleq 2814. Note that ax-ext 2793 is also a direct consequence of dfcleq 2815 (as an instance of its forward implication). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremeqriv 2818* Infer equality of classes from equivalence of membership. (Contributed by NM, 21-Jun-1993.)
(𝑥𝐴𝑥𝐵)       𝐴 = 𝐵

Theoremeqrdv 2819* Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.)
(𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)

Theoremeqrdav 2820* Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
((𝜑𝑥𝐴) → 𝑥𝐶)    &   ((𝜑𝑥𝐵) → 𝑥𝐶)    &   ((𝜑𝑥𝐶) → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)

Theoremeqid 2821 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also biid 264. An early mention of this law can be found in Aristotle, Metaphysics, Z.17, 1041a10-20. (Thanks to Stefan Allan and BJ for this information.) (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Oct-2017.)

𝐴 = 𝐴

Theoremeqidd 2822 Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
(𝜑𝐴 = 𝐴)

Theoremeqeq1d 2823 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐶))

Theoremeqeq1dALT 2824 Shorter proof of eqeq1d 2823 based on more axioms. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 19-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐶))

Theoremeqeq1 2825 Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))

Theoremeqeq1i 2826 Inference from equality to equivalence of equalities. (Contributed by NM, 15-Jul-1993.)
𝐴 = 𝐵       (𝐴 = 𝐶𝐵 = 𝐶)

Theoremeqcomd 2827 Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) Allow shortening of eqcom 2828. (Revised by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑𝐵 = 𝐴)

Theoremeqcom 2828 Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝐴 = 𝐵𝐵 = 𝐴)

Theoremeqcoms 2829 Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 24-Jun-1993.)
(𝐴 = 𝐵𝜑)       (𝐵 = 𝐴𝜑)

Theoremeqcomi 2830 Inference from commutative law for class equality. (Contributed by NM, 26-May-1993.)
𝐴 = 𝐵       𝐵 = 𝐴

Theoremneqcomd 2831 Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑 → ¬ 𝐵 = 𝐴)

Theoremeqeq2d 2832 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Allow shortening of eqeq2 2833. (Revised by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶 = 𝐴𝐶 = 𝐵))

Theoremeqeq2 2833 Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝐴 = 𝐵 → (𝐶 = 𝐴𝐶 = 𝐵))

Theoremeqeq2i 2834 Inference from equality to equivalence of equalities. (Contributed by NM, 26-May-1993.)
𝐴 = 𝐵       (𝐶 = 𝐴𝐶 = 𝐵)

Theoremeqeq12 2835 Equality relationship among four classes. (Contributed by NM, 3-Aug-1994.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremeqeq12i 2836 A useful inference for substituting definitions into an equality. (Contributed by NM, 15-Jul-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴 = 𝐶𝐵 = 𝐷)

Theoremeqeq12d 2837 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremeqeqan12d 2838 A useful inference for substituting definitions into an equality. See also eqeqan12dALT 2839. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremeqeqan12dALT 2839 Alternate proof of eqeqan12d 2838. This proof has one more step but one fewer essential step. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremeqeqan12rd 2840 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremeqtr 2841 Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)
((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)

Theoremeqtr2 2842 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
((𝐴 = 𝐵𝐴 = 𝐶) → 𝐵 = 𝐶)

Theoremeqtr3 2843 A transitive law for class equality. (Contributed by NM, 20-May-2005.)
((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)

Theoremeqtri 2844 An equality transitivity inference. (Contributed by NM, 26-May-1993.)
𝐴 = 𝐵    &   𝐵 = 𝐶       𝐴 = 𝐶

Theoremeqtr2i 2845 An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
𝐴 = 𝐵    &   𝐵 = 𝐶       𝐶 = 𝐴

Theoremeqtr3i 2846 An equality transitivity inference. (Contributed by NM, 6-May-1994.)
𝐴 = 𝐵    &   𝐴 = 𝐶       𝐵 = 𝐶

Theoremeqtr4i 2847 An equality transitivity inference. (Contributed by NM, 26-May-1993.)
𝐴 = 𝐵    &   𝐶 = 𝐵       𝐴 = 𝐶

Theorem3eqtri 2848 An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
𝐴 = 𝐵    &   𝐵 = 𝐶    &   𝐶 = 𝐷       𝐴 = 𝐷

Theorem3eqtrri 2849 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
𝐴 = 𝐵    &   𝐵 = 𝐶    &   𝐶 = 𝐷       𝐷 = 𝐴

Theorem3eqtr2i 2850 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
𝐴 = 𝐵    &   𝐶 = 𝐵    &   𝐶 = 𝐷       𝐴 = 𝐷

Theorem3eqtr2ri 2851 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
𝐴 = 𝐵    &   𝐶 = 𝐵    &   𝐶 = 𝐷       𝐷 = 𝐴

Theorem3eqtr3i 2852 An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
𝐴 = 𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶 = 𝐷

Theorem3eqtr3ri 2853 An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
𝐴 = 𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐷 = 𝐶

Theorem3eqtr4i 2854 An inference from three chained equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
𝐴 = 𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶 = 𝐷

Theorem3eqtr4ri 2855 An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
𝐴 = 𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐷 = 𝐶

Theoremeqtrd 2856 An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)

Theoremeqtr2d 2857 An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐶 = 𝐴)

Theoremeqtr3d 2858 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴 = 𝐶)       (𝜑𝐵 = 𝐶)

Theoremeqtr4d 2859 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴 = 𝐶)

Theorem3eqtrd 2860 A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐴 = 𝐷)

Theorem3eqtrrd 2861 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐷 = 𝐴)

Theorem3eqtr2d 2862 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐴 = 𝐷)

Theorem3eqtr2rd 2863 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑𝐷 = 𝐴)

Theorem3eqtr3d 2864 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶 = 𝐷)

Theorem3eqtr3rd 2865 A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐷 = 𝐶)

Theorem3eqtr4d 2866 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶 = 𝐷)

Theorem3eqtr4rd 2867 A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐷 = 𝐶)

Theoremsyl5eq 2868 An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
𝐴 = 𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)

Theoremsyl5req 2869 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
𝐴 = 𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐶 = 𝐴)

Theoremsyl5eqr 2870 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
𝐵 = 𝐴    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)

Theoremsyl5reqr 2871 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
𝐵 = 𝐴    &   (𝜑𝐵 = 𝐶)       (𝜑𝐶 = 𝐴)

Theoremsyl6eq 2872 An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
(𝜑𝐴 = 𝐵)    &   𝐵 = 𝐶       (𝜑𝐴 = 𝐶)

Theoremsyl6req 2873 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)    &   𝐵 = 𝐶       (𝜑𝐶 = 𝐴)

Theoremsyl6eqr 2874 An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
(𝜑𝐴 = 𝐵)    &   𝐶 = 𝐵       (𝜑𝐴 = 𝐶)

Theoremsyl6reqr 2875 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)    &   𝐶 = 𝐵       (𝜑𝐶 = 𝐴)

Theoremsylan9eq 2876 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐵 = 𝐶)       ((𝜑𝜓) → 𝐴 = 𝐶)

Theoremsylan9req 2877 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
(𝜑𝐵 = 𝐴)    &   (𝜓𝐵 = 𝐶)       ((𝜑𝜓) → 𝐴 = 𝐶)

Theoremsylan9eqr 2878 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐵 = 𝐶)       ((𝜓𝜑) → 𝐴 = 𝐶)

Theorem3eqtr3g 2879 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
(𝜑𝐴 = 𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶 = 𝐷)

Theorem3eqtr3a 2880 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
𝐴 = 𝐵    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶 = 𝐷)

Theorem3eqtr4g 2881 A chained equality inference, useful for converting to definitions. (Contributed by NM, 21-Jun-1993.)
(𝜑𝐴 = 𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶 = 𝐷)

Theorem3eqtr4a 2882 A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
𝐴 = 𝐵    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶 = 𝐷)

Theoremeq2tri 2883 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
(𝐴 = 𝐶𝐷 = 𝐹)    &   (𝐵 = 𝐷𝐶 = 𝐺)       ((𝐴 = 𝐶𝐵 = 𝐹) ↔ (𝐵 = 𝐷𝐴 = 𝐺))

Theoremabbi1 2884 Equivalent formulas yield equal class abstractions (closed form). This is the forward implication of abbi 2888, proved from fewer axioms. (Contributed by BJ and WL and SN, 20-Aug-2023.)
(∀𝑥(𝜑𝜓) → {𝑥𝜑} = {𝑥𝜓})

Theoremabbidv 2885* Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.) Avoid ax-12 2178, based on an idea of Steven Nguyen. (Revised by Wolf Lammen, 6-May-2023.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Theoremabbii 2886 Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 26-May-1993.) Remove dependency on ax-10 2146, ax-11 2162, and ax-12 2178. (Revised by Steven Nguyen, 3-May-2023.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}

Theoremabbid 2887 Equivalent wff's yield equal class abstractions (deduction form, with nonfreeness hypothesis). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) Avoid ax-10 2146 and ax-11 2162. (Revised by Wolf Lammen, 6-May-2023.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Theoremabbi 2888 Equivalent formulas define equal class abstractions, and conversely. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-8 2117 and df-clel 2892 (by avoiding use of cleqh 2935). (Revised by BJ, 23-Jun-2019.)
(∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})

Theoremcbvabv 2889* Rule used to change bound variables, using implicit substitution. Version of cbvab 2891 with disjoint variable conditions requiring fewer axioms. (Contributed by NM, 26-May-1999.) Require 𝑥, 𝑦 be disjoint to avoid ax-11 2162 and ax-13 2391. (Revised by Steven Nguyen, 4-Dec-2022.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

Theoremcbvabw 2890* Rule used to change bound variables, using implicit substitution. Version of cbvab 2891 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

Theoremcbvab 2891 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2391. Usage of the weaker cbvabw 2890 and cbvabv 2889 are preferred. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

2.1.2.3  Class membership

Definitiondf-clel 2892* Define the membership connective between classes. Theorem 6.3 of [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we adopt as a definition. See these references for its metalogical justification.

The hypotheses express that all instances of the conclusion where class variables are replaced with setvar variables hold. Therefore, this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This is only a proof sketch of conservativity; for details see Appendix of [Levy] p. 357. This is the reason why we call this axiomatic statement a "definition", even though it does not have the usual form of a definition. If we required a definition to have the usual form, we would call df-clel 2892 an axiom.

See also comments under df-clab 2800, df-cleq 2814, and abeq2 2944.

Alternate characterizations of 𝐴𝐵 when either 𝐴 or 𝐵 is a set are given by clel2 3630, clel3 3632, and clel4 3633.

This is called the "axiom of membership" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms.

While the three class definitions df-clab 2800, df-cleq 2814, and df-clel 2892 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker.

For a general discussion of the theory of classes, see mmset.html#class 2892. (Contributed by NM, 26-May-1993.) (Revised by BJ, 27-Jun-2019.)

(𝑦𝑧 ↔ ∃𝑢(𝑢 = 𝑦𝑢𝑧))    &   (𝑡𝑡 ↔ ∃𝑣(𝑣 = 𝑡𝑣𝑡))       (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))

Theoremdfclel 2893* Characterization of the elements of a class. (Contributed by BJ, 27-Jun-2019.)
(𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))

Theoremeleq1w 2894 Weaker version of eleq1 2899 (but more general than elequ1 2122) not depending on ax-ext 2793 nor df-cleq 2814.

Note that this provides a proof of ax-8 2117 from Tarski's FOL and dfclel 2893 (simply consider an instance where 𝐴 is replaced by a setvar and deduce the forward implication by biimpd 232), which shows that dfclel 2893 is too powerful to be used as a definition instead of df-clel 2892. (Contributed by BJ, 24-Jun-2019.)

(𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))

Theoremeleq2w 2895 Weaker version of eleq2 2900 (but more general than elequ2 2130) not depending on ax-ext 2793 nor df-cleq 2814. (Contributed by BJ, 29-Sep-2019.)
(𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))

Theoremeleq1d 2896 Deduction from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993.) Allow shortening of eleq1 2899. (Revised by Wolf Lammen, 20-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))

Theoremeleq2d 2897 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theoremeleq2dALT 2898 Alternate proof of eleq2d 2897, shorter at the expense of requiring ax-12 2178. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 20-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theoremeleq1 2899 Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theoremeleq2 2900 Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45093
 Copyright terms: Public domain < Previous  Next >