Definition df-ihash 10953

Description: Define the set size function ♯, which gives the cardinality of a finite set as a member of NN0, and assigns all infinite sets the value +oo. For example, (♯ ` { 0 , 1 , 2 } ) = 3.

Since we don't know that an arbitrary set is either finite or infinite (by inffiexmid 7024), the behavior beyond finite sets is not as useful as it might appear. For example, we wouldn't expect to be able to define this function in a meaningful way on ~P 1o, which cannot be shown to be finite (per pw1fin 7028).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8685). We adopt the former notation from Corollary 8.2.4 of [AczelRathjen], p. 80 (although that work only defines it for finite sets).

This definition (in terms of U. and ~<_) is not taken directly from the literature, but for finite sets should be equivalent to the conventional definition that the size of a finite set is the unique natural number which is equinumerous to the given set. (Contributed by Jim Kingdon, 19-Feb-2022.)

Assertion

Ref	Expression
df-ihash	|- ♯ = ( (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. om , +oo >. } ) o. ( x e. _V |-> U. { y e. ( om u. { om } ) | y ~<_ x } ) )

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-ihash

Step	Hyp	Ref	Expression
1		chash 10952	. 2 class ♯
2		vx	. . . . . 6 setvar x
3		cz 9402	. . . . . 6 class ZZ
4	2	cv 1372	. . . . . . 7 class x
5		c1 7956	. . . . . . 7 class 1
6		caddc 7958	. . . . . . 7 class +
7	4, 5, 6	co 5962	. . . . . 6 class ( x + 1 )
8	2, 3, 7	cmpt 4116	. . . . 5 class ( x e. ZZ |-> ( x + 1 ) )
9		cc0 7955	. . . . 5 class 0
10	8, 9	cfrec 6494	. . . 4 class frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
11		com 4651	. . . . . 6 class om
12		cpnf 8134	. . . . . 6 class +oo
13	11, 12	cop 3641	. . . . 5 class <. om , +oo >.
14	13	csn 3638	. . . 4 class { <. om , +oo >. }
15	10, 14	cun 3168	. . 3 class (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. om , +oo >. } )
16		cvv 2773	. . . 4 class _V
17		vy	. . . . . . . 8 setvar y
18	17	cv 1372	. . . . . . 7 class y
19		cdom 6844	. . . . . . 7 class ~<_
20	18, 4, 19	wbr 4054	. . . . . 6 wff y ~<_ x
21	11	csn 3638	. . . . . . 7 class { om }
22	11, 21	cun 3168	. . . . . 6 class ( om u. { om } )
23	20, 17, 22	crab 2489	. . . . 5 class { y e. ( om u. { om } ) | y ~<_ x }
24	23	cuni 3859	. . . 4 class U. { y e. ( om u. { om } ) | y ~<_ x }
25	2, 16, 24	cmpt 4116	. . 3 class ( x e. _V |-> U. { y e. ( om u. { om } ) | y ~<_ x } )
26	15, 25	ccom 4692	. 2 class ( (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. om , +oo >. } ) o. ( x e. _V |-> U. { y e. ( om u. { om } ) | y ~<_ x } ) )
27	1, 26	wceq 1373	1 wff ♯ = ( (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. om , +oo >. } ) o. ( x e. _V |-> U. { y e. ( om u. { om } ) | y ~<_ x } ) )

This definition is referenced by:  hashinfom  10955  hashennn  10957