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Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsyl5eleq 2201 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eleqr 2202 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eqel 2203 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eqelr 2204 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eleq 2205 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eleqr 2206 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)

Theoremeleq2s 2207 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremeqneltrd 2208 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.)

Theoremeqneltrrd 2209 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.)

Theoremneleqtrd 2210 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.)

Theoremneleqtrrd 2211 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.)

Theoremcleqh 2212* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2277. (Contributed by NM, 5-Aug-1993.)

Theoremnelneq 2213 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)

Theoremnelneq2 2214 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)

Theoremeqsb3lem 2215* Lemma for eqsb3 2216. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremeqsb3 2216* Substitution applied to an atomic wff (class version of equsb3 1898). (Contributed by Rodolfo Medina, 28-Apr-2010.)

Theoremclelsb3 2217* Substitution applied to an atomic wff (class version of elsb3 1925). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremclelsb4 2218* Substitution applied to an atomic wff (class version of elsb4 1926). (Contributed by Jim Kingdon, 22-Nov-2018.)

Theoremhbxfreq 2219 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1429 for equivalence version. (Contributed by NM, 21-Aug-2007.)

Theoremhblem 2220* Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremabeq2 2221* 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 2226 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 (that has a free variable ) to a theorem with a class variable , we substitute for throughout and simplify, where is a new class variable not already in the wff. Conversely, to convert a theorem with a class variable to one with , we substitute for throughout and simplify, where and 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.)

Theoremabeq1 2222* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)

Theoremabeq2i 2223 Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.)

Theoremabeq1i 2224 Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.)

Theoremabeq2d 2225 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)

Theoremabbi 2226 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)

Theoremabbi2i 2227* Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.)

Theoremabbii 2228 Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.)

Theoremabbid 2229 Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremabbidv 2230* Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.)

Theoremabbi2dv 2231* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Theoremabbi1dv 2232* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Theoremabid2 2233* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)

Theoremsb8ab 2234 Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)

Theoremcbvab 2235 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremcbvabv 2236* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)

Theoremclelab 2237* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)

Theoremclabel 2238* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)

Theoremsbab 2239* The right-hand side of the second equality is a way of representing proper substitution of for into a class variable. (Contributed by NM, 14-Sep-2003.)

2.1.3  Class form not-free predicate

Syntaxwnfc 2240 Extend wff definition to include the not-free predicate for classes.

Theoremnfcjust 2241* Justification theorem for df-nfc 2242. (Contributed by Mario Carneiro, 13-Oct-2016.)

Definitiondf-nfc 2242* Define the not-free predicate for classes. This is read " is not free in ". Not-free means that the value of cannot affect the value of , e.g., any occurrence of in 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 1418 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfci 2243* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcii 2244* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcr 2245* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcrii 2246* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcri 2247* Consequence of the not-free predicate. (Note that unlike nfcr 2245, this does not require and to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcd 2248* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfceqi 2249 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcxfr 2250 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcxfrd 2251 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfceqdf 2252 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremnfcv 2253* If is disjoint from , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcvd 2254* If is disjoint from , then is not free in . (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremnfab1 2255 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfnfc1 2256 is bound in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremclelsb3f 2257 Substitution applied to an atomic wff (class version of elsb3 1925). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)

Theoremnfab 2258 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfaba1 2259 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremnfnfc 2260 Hypothesis builder for . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfeq 2261 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfel 2262 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfeq1 2263* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)

Theoremnfel1 2264* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)

Theoremnfeq2 2265* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)

Theoremnfel2 2266* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)

Theoremnfcrd 2267* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfeqd 2268 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremnfeld 2269 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremdrnfc1 2270 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdrnfc2 2271 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfabd 2272 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdvelimdc 2273 Deduction form of dvelimc 2274. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdvelimc 2274 Version of dvelim 1966 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfcvf 2275 If and are distinct, then is not free in . (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfcvf2 2276 If and are distinct, then is not free in . (Contributed by Mario Carneiro, 5-Dec-2016.)

Theoremcleqf 2277 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2212. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremabid2f 2278 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.)

Theoremsbabel 2279* 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.)

2.1.4  Negated equality and membership

2.1.4.1  Negated equality

Syntaxwne 2280 Extend wff notation to include inequality.

Definitiondf-ne 2281 Define inequality. (Contributed by NM, 5-Aug-1993.)

Theoremneii 2282 Inference associated with df-ne 2281. (Contributed by BJ, 7-Jul-2018.)

Theoremneir 2283 Inference associated with df-ne 2281. (Contributed by BJ, 7-Jul-2018.)

Theoremnner 2284 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)

Theoremnnedc 2285 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
DECID

Theoremdcned 2286 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
DECID        DECID

Theoremneqned 2287 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2301. One-way deduction form of df-ne 2281. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2330. (Revised by Wolf Lammen, 22-Nov-2019.)

Theoremneqne 2288 From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremneirr 2289 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremeqneqall 2290 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)

Theoremdcne 2291 Decidable equality expressed in terms of . Basically the same as df-dc 803. (Contributed by Jim Kingdon, 14-Mar-2020.)
DECID

Theoremnonconne 2292 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Theoremneeq1 2293 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq2 2294 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq1i 2295 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq2i 2296 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq12i 2297 Inference for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneeq1d 2298 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq2d 2299 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq12d 2300 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)

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