Definition df-ihash 11139

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 7166), 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 7170).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8856). 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 11138	. 2 class ♯
2		vx	. . . . . 6 setvar x
3		cz 9577	. . . . . 6 class ZZ
4	2	cv 1397	. . . . . . 7 class x
5		c1 8128	. . . . . . 7 class 1
6		caddc 8130	. . . . . . 7 class +
7	4, 5, 6	co 6050	. . . . . 6 class ( x + 1 )
8	2, 3, 7	cmpt 4171	. . . . 5 class ( x e. ZZ |-> ( x + 1 ) )
9		cc0 8127	. . . . 5 class 0
10	8, 9	cfrec 6621	. . . 4 class frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
11		com 4712	. . . . . 6 class om
12		cpnf 8305	. . . . . 6 class +oo
13	11, 12	cop 3692	. . . . 5 class <. om , +oo >.
14	13	csn 3689	. . . 4 class { <. om , +oo >. }
15	10, 14	cun 3209	. . 3 class (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. om , +oo >. } )
16		cvv 2813	. . . 4 class _V
17		vy	. . . . . . . 8 setvar y
18	17	cv 1397	. . . . . . 7 class y
19		cdom 6974	. . . . . . 7 class ~<_
20	18, 4, 19	wbr 4109	. . . . . 6 wff y ~<_ x
21	11	csn 3689	. . . . . . 7 class { om }
22	11, 21	cun 3209	. . . . . 6 class ( om u. { om } )
23	20, 17, 22	crab 2524	. . . . 5 class { y e. ( om u. { om } ) | y ~<_ x }
24	23	cuni 3914	. . . 4 class U. { y e. ( om u. { om } ) | y ~<_ x }
25	2, 16, 24	cmpt 4171	. . 3 class ( x e. _V |-> U. { y e. ( om u. { om } ) | y ~<_ x } )
26	15, 25	ccom 4753	. 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 1398	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  11141  hashennn  11143