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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))
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