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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nelneq2 2201 | A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶) | ||
Theorem | eqsb3lem 2202* | Lemma for eqsb3 2203. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴) | ||
Theorem | eqsb3 2203* | Substitution applied to an atomic wff (class version of equsb3 1885). (Contributed by Rodolfo Medina, 28-Apr-2010.) |
⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴) | ||
Theorem | clelsb3 2204* | Substitution applied to an atomic wff (class version of elsb3 1912). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | ||
Theorem | clelsb4 2205* | Substitution applied to an atomic wff (class version of elsb4 1913). (Contributed by Jim Kingdon, 22-Nov-2018.) |
⊢ ([𝑥 / 𝑦]𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥) | ||
Theorem | hbxfreq 2206 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1416 for equivalence version. (Contributed by NM, 21-Aug-2007.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) ⇒ ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | ||
Theorem | hblem 2207* | 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 2208* |
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 2213 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 2209* | Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.) |
⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) | ||
Theorem | abeq2i 2210 | 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 2211 | 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 2212 | 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 2213 | 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 2214* | 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 2215 | Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} | ||
Theorem | abbid 2216 | 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 2217* | 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 2218* | Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) ⇒ ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) | ||
Theorem | abbi1dv 2219* | Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) |
⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴) | ||
Theorem | abid2 2220* | A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.) |
⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | ||
Theorem | sb8ab 2221 | Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑} | ||
Theorem | cbvab 2222 | 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 2223* | 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 2224* | Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) |
⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | ||
Theorem | clabel 2225* | 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 2226* | 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 2227 | Extend wff definition to include the not-free predicate for classes. |
wff Ⅎ𝑥𝐴 | ||
Theorem | nfcjust 2228* | Justification theorem for df-nfc 2229. (Contributed by Mario Carneiro, 13-Oct-2016.) |
⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) | ||
Definition | df-nfc 2229* | 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 1405 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfci 2230* | 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 2231* | 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 2232* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfcrii 2233* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfcri 2234* | Consequence of the not-free predicate. (Note that unlike nfcr 2232, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | ||
Theorem | nfcd 2235* | 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 2236 | Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) | ||
Theorem | nfcxfr 2237 | 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 2238 | 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 2239 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) | ||
Theorem | nfcv 2240* | If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 | ||
Theorem | nfcvd 2241* | If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) | ||
Theorem | nfab1 2242 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | ||
Theorem | nfnfc1 2243 | 𝑥 is bound in Ⅎ𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | ||
Theorem | clelsb3f 2244 | Substitution applied to an atomic wff (class version of elsb3 1912). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | ||
Theorem | nfab 2245 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑} | ||
Theorem | nfaba1 2246 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} | ||
Theorem | nfnfc 2247 | Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥Ⅎ𝑦𝐴 | ||
Theorem | nfeq 2248 | Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel 2249 | Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfeq1 2250* | Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel1 2251* | Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfeq2 2252* | Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 = 𝐵 | ||
Theorem | nfel2 2253* | Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | ||
Theorem | nfcrd 2254* | Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) | ||
Theorem | nfeqd 2255 | Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) | ||
Theorem | nfeld 2256 | Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) | ||
Theorem | drnfc1 2257 | 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 2258 | 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)) | ||
Theorem | nfabd 2259 | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | ||
Theorem | dvelimdc 2260 | Deduction form of dvelimc 2261. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑧𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑧𝐵) & ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) | ||
Theorem | dvelimc 2261 | Version of dvelim 1953 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑧𝐵 & ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵) | ||
Theorem | nfcvf 2262 | If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | ||
Theorem | nfcvf2 2263 | If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) | ||
Theorem | cleqf 2264 | Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2199. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | abid2f 2265 | A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | ||
Theorem | sbabel 2266* | Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴) | ||
Syntax | wne 2267 | Extend wff notation to include inequality. |
wff 𝐴 ≠ 𝐵 | ||
Definition | df-ne 2268 | Define inequality. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | ||
Theorem | neii 2269 | Inference associated with df-ne 2268. (Contributed by BJ, 7-Jul-2018.) |
⊢ 𝐴 ≠ 𝐵 ⇒ ⊢ ¬ 𝐴 = 𝐵 | ||
Theorem | neir 2270 | Inference associated with df-ne 2268. (Contributed by BJ, 7-Jul-2018.) |
⊢ ¬ 𝐴 = 𝐵 ⇒ ⊢ 𝐴 ≠ 𝐵 | ||
Theorem | nner 2271 | Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.) |
⊢ (𝐴 = 𝐵 → ¬ 𝐴 ≠ 𝐵) | ||
Theorem | nnedc 2272 | Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.) |
⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | dcned 2273 | Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.) |
⊢ (𝜑 → DECID 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → DECID 𝐴 ≠ 𝐵) | ||
Theorem | neqned 2274 | If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2288. One-way deduction form of df-ne 2268. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2317. (Revised by Wolf Lammen, 22-Nov-2019.) |
⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | neqne 2275 | From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | ||
Theorem | neirr 2276 | No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.) |
⊢ ¬ 𝐴 ≠ 𝐴 | ||
Theorem | eqneqall 2277 | A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑)) | ||
Theorem | dcne 2278 | Decidable equality expressed in terms of ≠. Basically the same as df-dc 787. (Contributed by Jim Kingdon, 14-Mar-2020.) |
⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | ||
Theorem | nonconne 2279 | Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) |
⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) | ||
Theorem | neeq1 2280 | Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) | ||
Theorem | neeq2 2281 | Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) | ||
Theorem | neeq1i 2282 | Inference for inequality. (Contributed by NM, 29-Apr-2005.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶) | ||
Theorem | neeq2i 2283 | Inference for inequality. (Contributed by NM, 29-Apr-2005.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵) | ||
Theorem | neeq12i 2284 | Inference for inequality. (Contributed by NM, 24-Jul-2012.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷) | ||
Theorem | neeq1d 2285 | Deduction for inequality. (Contributed by NM, 25-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) | ||
Theorem | neeq2d 2286 | Deduction for inequality. (Contributed by NM, 25-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) | ||
Theorem | neeq12d 2287 | Deduction for inequality. (Contributed by NM, 24-Jul-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)) | ||
Theorem | neneqd 2288 | Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | ||
Theorem | neneq 2289 | From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵) | ||
Theorem | eqnetri 2290 | Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 ≠ 𝐶 ⇒ ⊢ 𝐴 ≠ 𝐶 | ||
Theorem | eqnetrd 2291 | Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐶) | ||
Theorem | eqnetrri 2292 | Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 ≠ 𝐶 ⇒ ⊢ 𝐵 ≠ 𝐶 | ||
Theorem | eqnetrrd 2293 | Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ≠ 𝐶) | ||
Theorem | neeqtri 2294 | Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
⊢ 𝐴 ≠ 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴 ≠ 𝐶 | ||
Theorem | neeqtrd 2295 | Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐶) | ||
Theorem | neeqtrri 2296 | Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
⊢ 𝐴 ≠ 𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴 ≠ 𝐶 | ||
Theorem | neeqtrrd 2297 | Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐶) | ||
Theorem | syl5eqner 2298 | B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐶) | ||
Theorem | 3netr3d 2299 | Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐶 ≠ 𝐷) | ||
Theorem | 3netr4d 2300 | Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐴) & ⊢ (𝜑 → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
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