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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | etasslt 32801* | A restatement of noeta 32749 using set less than. (Contributed by Scott Fenton, 10-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))) | ||
Theorem | scutbdaybnd 32802 | An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) | ||
Theorem | scutbdaylt 32803 | If a surreal lies in a gap and is not equal to the cut, its birthday is greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.) |
⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘𝑋)) | ||
Theorem | slerec 32804* | A comparison law for surreals considered as cuts of sets of surreals. In Conway's treatment, this is the definition of less than or equal. (Contributed by Scott Fenton, 11-Dec-2021.) |
⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌))) | ||
Theorem | sltrec 32805* | A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.) |
⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))) | ||
Syntax | cmade 32806 | Declare the symbol for the made by function. |
class M | ||
Syntax | cold 32807 | Declare the symbol for the older than function. |
class O | ||
Syntax | cnew 32808 | Declare the symbol for the new on function. |
class N | ||
Syntax | cleft 32809 | Declare the symbol for the left option function. |
class L | ||
Syntax | cright 32810 | Declare the symbol for the right option function. |
class R | ||
Definition | df-made 32811 | Define the made by function. This function carries an ordinal to all surreals made by sections of surreals older than it. (Contributed by Scott Fenton, 17-Dec-2021.) |
⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓)))) | ||
Definition | df-old 32812 | Define the older than function. This function carries an ordinal to all surreals made by a previous ordinal. (Contributed by Scott Fenton, 17-Dec-2021.) |
⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | ||
Definition | df-new 32813 | Define the newer than function. This function carries an ordinal to all surreals made on that day. (Contributed by Scott Fenton, 17-Dec-2021.) |
⊢ N = (𝑥 ∈ On ↦ (( O ‘𝑥) ∖ ( M ‘𝑥))) | ||
Definition | df-left 32814* | Define the left options of a surreal. This is the set of surreals that are "closest" on the left to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.) |
⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ ∀𝑧 ∈ No ((𝑦 <s 𝑧 ∧ 𝑧 <s 𝑥) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))}) | ||
Definition | df-right 32815* | Define the left options of a surreal. This is the set of surreals that are "closest" on the right to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.) |
⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))}) | ||
Theorem | madeval 32816 | The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.) |
⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) | ||
Theorem | madeval2 32817* | Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.) |
⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)}) | ||
Syntax | ctxp 32818 | Declare the syntax for tail Cartesian product. |
class (𝐴 ⊗ 𝐵) | ||
Syntax | cpprod 32819 | Declare the syntax for the parallel product. |
class pprod(𝑅, 𝑆) | ||
Syntax | csset 32820 | Declare the subset relationship class. |
class SSet | ||
Syntax | ctrans 32821 | Declare the transitive set class. |
class Trans | ||
Syntax | cbigcup 32822 | Declare the set union relationship. |
class Bigcup | ||
Syntax | cfix 32823 | Declare the syntax for the fixpoints of a class. |
class Fix 𝐴 | ||
Syntax | climits 32824 | Declare the class of limit ordinals. |
class Limits | ||
Syntax | cfuns 32825 | Declare the syntax for the class of all function. |
class Funs | ||
Syntax | csingle 32826 | Declare the syntax for the singleton function. |
class Singleton | ||
Syntax | csingles 32827 | Declare the syntax for the class of all singletons. |
class Singletons | ||
Syntax | cimage 32828 | Declare the syntax for the image functor. |
class Image𝐴 | ||
Syntax | ccart 32829 | Declare the syntax for the cartesian function. |
class Cart | ||
Syntax | cimg 32830 | Declare the syntax for the image function. |
class Img | ||
Syntax | cdomain 32831 | Declare the syntax for the domain function. |
class Domain | ||
Syntax | crange 32832 | Declare the syntax for the range function. |
class Range | ||
Syntax | capply 32833 | Declare the syntax for the application function. |
class Apply | ||
Syntax | ccup 32834 | Declare the syntax for the cup function. |
class Cup | ||
Syntax | ccap 32835 | Declare the syntax for the cap function. |
class Cap | ||
Syntax | csuccf 32836 | Declare the syntax for the successor function. |
class Succ | ||
Syntax | cfunpart 32837 | Declare the syntax for the functional part functor. |
class Funpart𝐹 | ||
Syntax | cfullfn 32838 | Declare the syntax for the full function functor. |
class FullFun𝐹 | ||
Syntax | crestrict 32839 | Declare the syntax for the restriction function. |
class Restrict | ||
Syntax | cub 32840 | Declare the syntax for the upper bound relationship functor. |
class UB𝑅 | ||
Syntax | clb 32841 | Declare the syntax for the lower bound relationship functor. |
class LB𝑅 | ||
Definition | df-txp 32842 | Define the tail cross of two classes. Membership in this class is defined by txpss3v 32866 and brtxp 32868. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) | ||
Definition | df-pprod 32843 | Define the parallel product of two classes. Membership in this class is defined by pprodss4v 32872 and brpprod 32873. (Contributed by Scott Fenton, 11-Apr-2014.) |
⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) | ||
Definition | df-sset 32844 | Define the subset class. For the value, see brsset 32877. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) | ||
Definition | df-trans 32845 | Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E )) | ||
Definition | df-bigcup 32846 | Define the Bigcup function, which, per fvbigcup 32890, carries a set to its union. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))) | ||
Definition | df-fix 32847 | Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Fix 𝐴 = dom (𝐴 ∩ I ) | ||
Definition | df-limits 32848 | Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅}) | ||
Definition | df-funs 32849 | Define the class of all functions. See elfuns 32903 for membership. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E ))) | ||
Definition | df-singleton 32850 | Define the singleton function. See brsingle 32905 for its value. (Contributed by Scott Fenton, 4-Apr-2014.) |
⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) | ||
Definition | df-singles 32851 | Define the class of all singletons. See elsingles 32906 for membership. (Contributed by Scott Fenton, 19-Feb-2013.) |
⊢ Singletons = ran Singleton | ||
Definition | df-image 32852 | Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴 “ 𝑥), providing that the latter exists. See imageval 32918 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.) |
⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) | ||
Definition | df-cart 32853 | Define the cartesian product function. See brcart 32920 for its value. (Contributed by Scott Fenton, 11-Apr-2014.) |
⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V))) | ||
Definition | df-img 32854 | Define the image function. See brimg 32925 for its value. (Contributed by Scott Fenton, 12-Apr-2014.) |
⊢ Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart) | ||
Definition | df-domain 32855 | Define the domain function. See brdomain 32921 for its value. (Contributed by Scott Fenton, 11-Apr-2014.) |
⊢ Domain = Image(1st ↾ (V × V)) | ||
Definition | df-range 32856 | Define the range function. See brrange 32922 for its value. (Contributed by Scott Fenton, 11-Apr-2014.) |
⊢ Range = Image(2nd ↾ (V × V)) | ||
Definition | df-cup 32857 | Define the little cup function. See brcup 32927 for its value. (Contributed by Scott Fenton, 14-Apr-2014.) |
⊢ Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∪ (◡2nd ∘ E )) ⊗ V))) | ||
Definition | df-cap 32858 | Define the little cap function. See brcap 32928 for its value. (Contributed by Scott Fenton, 17-Apr-2014.) |
⊢ Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∩ (◡2nd ∘ E )) ⊗ V))) | ||
Definition | df-restrict 32859 | Define the restriction function. See brrestrict 32937 for its value. (Contributed by Scott Fenton, 17-Apr-2014.) |
⊢ Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st ))))) | ||
Definition | df-succf 32860 | Define the successor function. See brsuccf 32929 for its value. (Contributed by Scott Fenton, 14-Apr-2014.) |
⊢ Succ = (Cup ∘ ( I ⊗ Singleton)) | ||
Definition | df-apply 32861 | Define the application function. See brapply 32926 for its value. (Contributed by Scott Fenton, 12-Apr-2014.) |
⊢ Apply = (( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))) | ||
Definition | df-funpart 32862 | Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 32931 and funpartfv 32933 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.) |
⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) | ||
Definition | df-fullfun 32863 | Define the full function over 𝐹. This is a function with domain V that always agrees with 𝐹 for its value. (Contributed by Scott Fenton, 17-Apr-2014.) |
⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) | ||
Definition | df-ub 32864 | Define the upper bound relationship functor. See brub 32942 for value. (Contributed by Scott Fenton, 3-May-2018.) |
⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E )) | ||
Definition | df-lb 32865 | Define the lower bound relationship functor. See brlb 32943 for value. (Contributed by Scott Fenton, 3-May-2018.) |
⊢ LB𝑅 = UB◡𝑅 | ||
Theorem | txpss3v 32866 | A tail Cartesian product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V)) | ||
Theorem | txprel 32867 | A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ Rel (𝐴 ⊗ 𝐵) | ||
Theorem | brtxp 32868 | Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 32866, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V & ⊢ 𝑍 ∈ V ⇒ ⊢ (𝑋(𝐴 ⊗ 𝐵)〈𝑌, 𝑍〉 ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍)) | ||
Theorem | brtxp2 32869* | The binary relation over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)) | ||
Theorem | dfpprod2 32870 | Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.) |
⊢ pprod(𝐴, 𝐵) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V))))) | ||
Theorem | pprodcnveq 32871 | A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.) |
⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆) | ||
Theorem | pprodss4v 32872 | The parallel product is a subclass of ((V × V) × (V × V)). (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V)) | ||
Theorem | brpprod 32873 | Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 32872, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V & ⊢ 𝑍 ∈ V & ⊢ 𝑊 ∈ V ⇒ ⊢ (〈𝑋, 𝑌〉pprod(𝐴, 𝐵)〈𝑍, 𝑊〉 ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊)) | ||
Theorem | brpprod3a 32874* | Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V & ⊢ 𝑍 ∈ V ⇒ ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) | ||
Theorem | brpprod3b 32875* | Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.) |
⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V & ⊢ 𝑍 ∈ V ⇒ ⊢ (𝑋pprod(𝑅, 𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) | ||
Theorem | relsset 32876 | The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ Rel SSet | ||
Theorem | brsset 32877 | For sets, the SSet binary relation is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 SSet 𝐵 ↔ 𝐴 ⊆ 𝐵) | ||
Theorem | idsset 32878 | I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ I = ( SSet ∩ ◡ SSet ) | ||
Theorem | eltrans 32879 | Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴) | ||
Theorem | dfon3 32880 | A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.) |
⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) | ||
Theorem | dfon4 32881 | Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.) |
⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) | ||
Theorem | brtxpsd 32882* | Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) | ||
Theorem | brtxpsd2 32883* | Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V))) & ⊢ 𝐴𝐶𝐵 ⇒ ⊢ (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) | ||
Theorem | brtxpsd3 32884* | A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V))) & ⊢ 𝐴𝐶𝐵 & ⊢ (𝑥 ∈ 𝑋 ↔ 𝑥𝑆𝐴) ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝐵 = 𝑋) | ||
Theorem | relbigcup 32885 | The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Rel Bigcup | ||
Theorem | brbigcup 32886 | Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵) | ||
Theorem | dfbigcup2 32887 | Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) | ||
Theorem | fobigcup 32888 | Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Bigcup :V–onto→V | ||
Theorem | fnbigcup 32889 | Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Bigcup Fn V | ||
Theorem | fvbigcup 32890 | For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ( Bigcup ‘𝐴) = ∪ 𝐴 | ||
Theorem | elfix 32891 | Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) | ||
Theorem | elfix2 32892 | Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Rel 𝑅 ⇒ ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) | ||
Theorem | dffix2 32893 | The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Fix 𝐴 = ran (𝐴 ∩ I ) | ||
Theorem | fixssdm 32894 | The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Fix 𝐴 ⊆ dom 𝐴 | ||
Theorem | fixssrn 32895 | The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Fix 𝐴 ⊆ ran 𝐴 | ||
Theorem | fixcnv 32896 | The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Fix 𝐴 = Fix ◡𝐴 | ||
Theorem | fixun 32897 | The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵) | ||
Theorem | ellimits 32898 | Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ Limits ↔ Lim 𝐴) | ||
Theorem | limitssson 32899 | The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Limits ⊆ On | ||
Theorem | dfom5b 32900 | A quantifier-free definition of ω that does not depend on ax-inf 8895. (Note: label was changed from dfom5 8907 to dfom5b 32900 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ ω = (On ∩ ∩ Limits ) |
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