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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | eleq1w 2201 | Weaker version of eleq1 2203 (but more general than elequ1 1691) not depending on ax-ext 2122 nor df-cleq 2133. (Contributed by BJ, 24-Jun-2019.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | ||
Theorem | eleq2w 2202 | Weaker version of eleq2 2204 (but more general than elequ2 1692) not depending on ax-ext 2122 nor df-cleq 2133. (Contributed by BJ, 29-Sep-2019.) |
⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) | ||
Theorem | eleq1 2203 | Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | ||
Theorem | eleq2 2204 | Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | ||
Theorem | eleq12 2205 | Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | ||
Theorem | eleq1i 2206 | Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶) | ||
Theorem | eleq2i 2207 | Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵) | ||
Theorem | eleq12i 2208 | Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷) | ||
Theorem | eleq1d 2209 | Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | ||
Theorem | eleq2d 2210 | Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | ||
Theorem | eleq12d 2211 | Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | ||
Theorem | eleq1a 2212 | A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.) |
⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵)) | ||
Theorem | eqeltri 2213 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ 𝐴 ∈ 𝐶 | ||
Theorem | eqeltrri 2214 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ 𝐵 ∈ 𝐶 | ||
Theorem | eleqtri 2215 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴 ∈ 𝐶 | ||
Theorem | eleqtrri 2216 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴 ∈ 𝐶 | ||
Theorem | eqeltrd 2217 | Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eqeltrrd 2218 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝐶) | ||
Theorem | eleqtrd 2219 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eleqtrrd 2220 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | 3eltr3i 2221 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐶 ∈ 𝐷 | ||
Theorem | 3eltr4i 2222 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ 𝐶 ∈ 𝐷 | ||
Theorem | 3eltr3d 2223 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) | ||
Theorem | 3eltr4d 2224 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐴) & ⊢ (𝜑 → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) | ||
Theorem | 3eltr3g 2225 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) | ||
Theorem | 3eltr4g 2226 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐷) | ||
Theorem | eqeltrid 2227 | B membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eqeltrrid 2228 | B membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eleqtrid 2229 | B membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eleqtrrid 2230 | B membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eqeltrdi 2231 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eqeltrrdi 2232 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eleqtrdi 2233 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eleqtrrdi 2234 | A membership and equality inference. (Contributed by NM, 24-Apr-2005.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eleq2s 2235 | Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝐴 ∈ 𝐵 → 𝜑) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝐴 ∈ 𝐶 → 𝜑) | ||
Theorem | eqneltrd 2236 | If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) | ||
Theorem | eqneltrrd 2237 | If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) | ||
Theorem | neleqtrd 2238 | If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) | ||
Theorem | neleqtrrd 2239 | If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) | ||
Theorem | cleqh 2240* | Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2306. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) & ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | nelneq 2241 | A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.) |
⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵) | ||
Theorem | nelneq2 2242 | A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶) | ||
Theorem | eqsb3lem 2243* | Lemma for eqsb3 2244. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴) | ||
Theorem | eqsb3 2244* | Substitution applied to an atomic wff (class version of equsb3 1925). (Contributed by Rodolfo Medina, 28-Apr-2010.) |
⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴) | ||
Theorem | clelsb3 2245* | Substitution applied to an atomic wff (class version of elsb3 1952). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | ||
Theorem | clelsb4 2246* | Substitution applied to an atomic wff (class version of elsb4 1953). (Contributed by Jim Kingdon, 22-Nov-2018.) |
⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦) | ||
Theorem | hbxfreq 2247 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1449 for equivalence version. (Contributed by NM, 21-Aug-2007.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) ⇒ ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | ||
Theorem | hblem 2248* | Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) |
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) ⇒ ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) | ||
Theorem | abeq2 2249* |
Equality of a class variable and a class abstraction (also called a
class builder). Theorem 5.1 of [Quine] p.
34. This theorem shows the
relationship between expressions with class abstractions and expressions
with class variables. Note that abbi 2254 and its relatives are among
those useful for converting theorems with class variables to equivalent
theorems with wff variables, by first substituting a class abstraction
for each class variable.
Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable 𝜑 (that has a free variable 𝑥) to a theorem with a class variable 𝐴, we substitute 𝑥 ∈ 𝐴 for 𝜑 throughout and simplify, where 𝐴 is a new class variable not already in the wff. Conversely, to convert a theorem with a class variable 𝐴 to one with 𝜑, we substitute {𝑥 ∣ 𝜑} for 𝐴 throughout and simplify, where 𝑥 and 𝜑 are new set and wff variables not already in the wff. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | ||
Theorem | abeq1 2250* | Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.) |
⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) | ||
Theorem | abeq2i 2251 | Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.) |
⊢ 𝐴 = {𝑥 ∣ 𝜑} ⇒ ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) | ||
Theorem | abeq1i 2252 | Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.) |
⊢ {𝑥 ∣ 𝜑} = 𝐴 ⇒ ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴) | ||
Theorem | abeq2d 2253 | Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.) |
⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) | ||
Theorem | abbi 2254 | Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) | ||
Theorem | abbi2i 2255* | Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) ⇒ ⊢ 𝐴 = {𝑥 ∣ 𝜑} | ||
Theorem | abbii 2256 | Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} | ||
Theorem | abbid 2257 | Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | ||
Theorem | abbidv 2258* | Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | ||
Theorem | abbi2dv 2259* | Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) ⇒ ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) | ||
Theorem | abbi1dv 2260* | Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) |
⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴) | ||
Theorem | abid2 2261* | A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.) |
⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | ||
Theorem | sb8ab 2262 | Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑} | ||
Theorem | cbvabw 2263* | Version of cbvab 2264 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | ||
Theorem | cbvab 2264 | Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | ||
Theorem | cbvabv 2265* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | ||
Theorem | clelab 2266* | Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) |
⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | ||
Theorem | clabel 2267* | 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 2268* | 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 2269 | Extend wff definition to include the not-free predicate for classes. |
wff Ⅎ𝑥𝐴 | ||
Theorem | nfcjust 2270* | Justification theorem for df-nfc 2271. (Contributed by Mario Carneiro, 13-Oct-2016.) |
⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) | ||
Definition | df-nfc 2271* | 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 quantifier or something that expands to one (such as "there exists at most one"). It is defined in terms of the not-free predicate df-nf 1438 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfci 2272* | 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 2273* | 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 2274* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfcrii 2275* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfcri 2276* | Consequence of the not-free predicate. (Note that unlike nfcr 2274, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | ||
Theorem | nfcd 2277* | 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 | nfceqi 2278 | Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) | ||
Theorem | nfcxfr 2279 | 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 2280 | 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 | nfceqdf 2281 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) | ||
Theorem | nfcv 2282* | If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 | ||
Theorem | nfcvd 2283* | If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) | ||
Theorem | nfab1 2284 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | ||
Theorem | nfnfc1 2285 | 𝑥 is bound in Ⅎ𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | ||
Theorem | clelsb3f 2286 | Substitution applied to an atomic wff (class version of elsb3 1952). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | ||
Theorem | nfab 2287 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑} | ||
Theorem | nfaba1 2288 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} | ||
Theorem | nfnfc 2289 | Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥Ⅎ𝑦𝐴 | ||
Theorem | nfeq 2290 | Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel 2291 | Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfeq1 2292* | Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel1 2293* | Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfeq2 2294* | Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel2 2295* | Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfcrd 2296* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfeqd 2297 | Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) | ||
Theorem | nfeld 2298 | Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) | ||
Theorem | drnfc1 2299 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵)) | ||
Theorem | drnfc2 2300 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵)) |
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