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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | eleq1ab 2801 |
Extension (in the sense of Remark 3 of the comment of df-clab 2800) of
elequ1 2112 from formulas of the form "setvar ∈ setvar" to formulas of
the form "setvar ∈ class
abstraction". This extension does not
require ax-8 2107 contrary to elequ1 2112, 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 2107.
This is an instance of eleq1w 2895 where the containing class is a class abstraction, and contrary to it, it can be proved without df-clel 2893. See also eleq1 2900 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 2893) was stressed by Mario Carneiro. (Contributed by BJ, 17-Aug-2023.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ 𝑦 ∈ {𝑧 ∣ 𝜑})) | ||
Theorem | cleljustab 2802* | Extension of cleljust 2114 from formulas of the form "setvar ∈ setvar" to formulas of the form "setvar ∈ class abstraction". This is an instance of dfclel 2894 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 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑})) | ||
Theorem | abid 2803 | Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 26-May-1993.) |
⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | ||
Theorem | vexwt 2804 | A standard theorem of predicate calculus (stdpc4 2064) expressed using class abstractions. Closed form of vexw 2805. (Contributed by BJ, 14-Jun-2019.) |
⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | vexw 2805 |
If 𝜑
is a theorem, then any set belongs to the class
{𝑥
∣ 𝜑}.
Therefore, {𝑥 ∣ 𝜑} is "a" universal class.
This is the closest one can get to defining a universal class, or proving vex 3498, without using ax-ext 2793. Note that this theorem has no disjoint variable condition and does not use df-clel 2893 nor df-cleq 2814 either: only propositional logic and ax-gen 1787 and df-clab 2800. This is sbt 2062 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 3496). 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 ∣ (𝜑 → 𝜑)}, {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 3497. Its degenerate instance is also a simple consequence of abid 2803 (using mpbir 232). (Contributed by BJ, 13-Jun-2019.) Reduce axiom dependencies. (Revised by Steven Nguyen, 25-Apr-2023.) |
⊢ 𝜑 ⇒ ⊢ 𝑦 ∈ {𝑥 ∣ 𝜑} | ||
Theorem | vextru 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 3497). This is sbtru 2063 expressed using class abstractions. (Contributed by BJ, 2-Sep-2023.) |
⊢ 𝑦 ∈ {𝑥 ∣ ⊤} | ||
Theorem | hbab1 2807* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.) |
⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | nfsab1 2808* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2167. (Revised by SN, 20-Sep-2023.) |
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | ||
Theorem | nfsab1OLD 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.) |
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | ||
Theorem | hbab 2810* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) Add disjoint variable condition to avoid ax-13 2383. See hbabg 2811 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) | ||
Theorem | hbabg 2811* | Bound-variable hypothesis builder for a class abstraction. See hbab 2810 for a version with more disjoint variable conditions, but not requiring ax-13 2383. (Contributed by NM, 1-Mar-1995.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) | ||
Theorem | nfsab 2812* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2383. See nfsabg 2813 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑} | ||
Theorem | nfsabg 2813* | Bound-variable hypothesis builder for a class abstraction. See nfsab 2812 for a version with more disjoint variable conditions, but not requiring ax-13 2383. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑} | ||
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 1528 and wcel 2105, 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). | ||
Definition | df-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 2893, and abeq2 2945. 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 2893 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 2893. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.) |
⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧)) & ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | dfcleq 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 1787, equid 2010 }. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.) |
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | cvjust 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 1527, 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 1527 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 2956 for the version of cvjust 2816 extended to classes. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2383. (Revised by Wolf Lammen, 4-May-2023.) |
⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} | ||
Theorem | ax9ALT 2817 | Proof of ax-9 2115 from Tarski's FOL and dfcleq 2815. For a version not using ax-8 2107 either, see bj-ax9 34114. 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 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | eqriv 2818* | Infer equality of classes from equivalence of membership. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ⇒ ⊢ 𝐴 = 𝐵 | ||
Theorem | eqrdv 2819* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | eqrdav 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.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | eqid 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 262. 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 | ||
Theorem | eqidd 2822 | Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) |
⊢ (𝜑 → 𝐴 = 𝐴) | ||
Theorem | eqeq1d 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.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | ||
Theorem | eqeq1dALT 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.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | ||
Theorem | eqeq1 2825 | Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | ||
Theorem | eqeq1i 2826 | Inference from equality to equivalence of equalities. (Contributed by NM, 15-Jul-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐶) | ||
Theorem | eqcomd 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.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐵 = 𝐴) | ||
Theorem | eqcom 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 = 𝐵 ↔ 𝐵 = 𝐴) | ||
Theorem | eqcoms 2829 | Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 24-Jun-1993.) |
⊢ (𝐴 = 𝐵 → 𝜑) ⇒ ⊢ (𝐵 = 𝐴 → 𝜑) | ||
Theorem | eqcomi 2830 | Inference from commutative law for class equality. (Contributed by NM, 26-May-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝐵 = 𝐴 | ||
Theorem | neqcomd 2831 | Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 = 𝐴) | ||
Theorem | eqeq2d 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.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)) | ||
Theorem | eqeq2 2833 | Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)) | ||
Theorem | eqeq2i 2834 | Inference from equality to equivalence of equalities. (Contributed by NM, 26-May-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵) | ||
Theorem | eqeq12 2835 | Equality relationship among four classes. (Contributed by NM, 3-Aug-1994.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqeq12i 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 = 𝐶 ↔ 𝐵 = 𝐷) | ||
Theorem | eqeq12d 2837 | A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqeqan12d 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.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqeqan12dALT 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.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqeqan12rd 2840 | A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqtr 2841 | Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶) | ||
Theorem | eqtr2 2842 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶) | ||
Theorem | eqtr3 2843 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) |
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵) | ||
Theorem | eqtri 2844 | An equality transitivity inference. (Contributed by NM, 26-May-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴 = 𝐶 | ||
Theorem | eqtr2i 2845 | An equality transitivity inference. (Contributed by NM, 21-Feb-1995.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐶 = 𝐴 | ||
Theorem | eqtr3i 2846 | An equality transitivity inference. (Contributed by NM, 6-May-1994.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 = 𝐶 ⇒ ⊢ 𝐵 = 𝐶 | ||
Theorem | eqtr4i 2847 | An equality transitivity inference. (Contributed by NM, 26-May-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴 = 𝐶 | ||
Theorem | 3eqtri 2848 | An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ 𝐴 = 𝐷 | ||
Theorem | 3eqtrri 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 | ||
Theorem | 3eqtr2i 2850 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ 𝐴 = 𝐷 | ||
Theorem | 3eqtr2ri 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 | ||
Theorem | 3eqtr3i 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 | ||
Theorem | 3eqtr3ri 2853 | An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐷 = 𝐶 | ||
Theorem | 3eqtr4i 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 | ||
Theorem | 3eqtr4ri 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 | ||
Theorem | eqtrd 2856 | An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | eqtr2d 2857 | An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | eqtr3d 2858 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
Theorem | eqtr4d 2859 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | 3eqtrd 2860 | A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → 𝐴 = 𝐷) | ||
Theorem | 3eqtrrd 2861 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → 𝐷 = 𝐴) | ||
Theorem | 3eqtr2d 2862 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → 𝐴 = 𝐷) | ||
Theorem | 3eqtr2rd 2863 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → 𝐷 = 𝐴) | ||
Theorem | 3eqtr3d 2864 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | 3eqtr3rd 2865 | A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐷 = 𝐶) | ||
Theorem | 3eqtr4d 2866 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐴) & ⊢ (𝜑 → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | 3eqtr4rd 2867 | A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐴) & ⊢ (𝜑 → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐷 = 𝐶) | ||
Theorem | syl5eq 2868 | An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl5req 2869 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | syl5eqr 2870 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl5reqr 2871 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | syl6eq 2872 | An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl6req 2873 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | syl6eqr 2874 | An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl6reqr 2875 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | sylan9eq 2876 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐵 = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) | ||
Theorem | sylan9req 2877 | An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ (𝜓 → 𝐵 = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) | ||
Theorem | sylan9eqr 2878 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐵 = 𝐶) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝐴 = 𝐶) | ||
Theorem | 3eqtr3g 2879 | A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | 3eqtr3a 2880 | A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | 3eqtr4g 2881 | A chained equality inference, useful for converting to definitions. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | 3eqtr4a 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 & ⊢ (𝜑 → 𝐶 = 𝐴) & ⊢ (𝜑 → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | eq2tri 2883 | A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.) |
⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹) & ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐺) ⇒ ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐹) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺)) | ||
Theorem | abbi1 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(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) | ||
Theorem | abbidv 2885* | Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.) Avoid ax-12 2167, based on an idea of Steven Nguyen. (Revised by Wolf Lammen, 6-May-2023.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | ||
Theorem | abbii 2886 | Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 26-May-1993.) Remove dependency on ax-10 2136, ax-11 2151, and ax-12 2167. (Revised by Steven Nguyen, 3-May-2023.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} | ||
Theorem | abbid 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 2136 and ax-11 2151. (Revised by Wolf Lammen, 6-May-2023.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | ||
Theorem | abbi 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 2107 and df-clel 2893 (by avoiding use of cleqh 2936). (Revised by BJ, 23-Jun-2019.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) | ||
Theorem | cbvabv 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 2151 and ax-13 2383. (Revised by Steven Nguyen, 4-Dec-2022.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | ||
Theorem | cbvabw 2890* | Version of cbvab 2891 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | ||
Theorem | cbvab 2891 | 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 2892* | Obsolete version of cbvabv 2889 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} | ||
Definition | df-clel 2893* |
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 2893 an axiom. See also comments under df-clab 2800, df-cleq 2814, and abeq2 2945. Alternate characterizations of 𝐴 ∈ 𝐵 when either 𝐴 or 𝐵 is a set are given by clel2 3652, clel3 3654, and clel4 3655. 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 2893 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 2893. (Contributed by NM, 26-May-1993.) (Revised by BJ, 27-Jun-2019.) |
⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧)) & ⊢ (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) | ||
Theorem | dfclel 2894* | Characterization of the elements of a class. (Contributed by BJ, 27-Jun-2019.) |
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) | ||
Theorem | eleq1w 2895 |
Weaker version of eleq1 2900 (but more general than elequ1 2112) not
depending on ax-ext 2793 nor df-cleq 2814.
Note that this provides a proof of ax-8 2107 from Tarski's FOL and dfclel 2894 (simply consider an instance where 𝐴 is replaced by a setvar and deduce the forward implication by biimpd 230), which shows that dfclel 2894 is too powerful to be used as a definition instead of df-clel 2893. (Contributed by BJ, 24-Jun-2019.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | ||
Theorem | eleq2w 2896 | Weaker version of eleq2 2901 (but more general than elequ2 2120) not depending on ax-ext 2793 nor df-cleq 2814. (Contributed by BJ, 29-Sep-2019.) |
⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) | ||
Theorem | eleq1d 2897 | Deduction from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993.) Allow shortening of eleq1 2900. (Revised by Wolf Lammen, 20-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | ||
Theorem | eleq2d 2898 | Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | ||
Theorem | eleq2dALT 2899 | Alternate proof of eleq2d 2898, shorter at the expense of requiring ax-12 2167. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 20-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | ||
Theorem | eleq1 2900 | Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) |
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