Definition df-ihash 11084

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 7141), 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 7145).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8804). 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 11083	. 2 class ♯
2		vx	. . . . . 6 setvar x
3		cz 9523	. . . . . 6 class ZZ
4	2	cv 1397	. . . . . . 7 class x
5		c1 8076	. . . . . . 7 class 1
6		caddc 8078	. . . . . . 7 class +
7	4, 5, 6	co 6028	. . . . . 6 class ( x + 1 )
8	2, 3, 7	cmpt 4155	. . . . 5 class ( x e. ZZ |-> ( x + 1 ) )
9		cc0 8075	. . . . 5 class 0
10	8, 9	cfrec 6599	. . . 4 class frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
11		com 4694	. . . . . 6 class om
12		cpnf 8253	. . . . . 6 class +oo
13	11, 12	cop 3676	. . . . 5 class <. om , +oo >.
14	13	csn 3673	. . . 4 class { <. om , +oo >. }
15	10, 14	cun 3199	. . 3 class (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. { <. om , +oo >. } )
16		cvv 2803	. . . 4 class _V
17		vy	. . . . . . . 8 setvar y
18	17	cv 1397	. . . . . . 7 class y
19		cdom 6951	. . . . . . 7 class ~<_
20	18, 4, 19	wbr 4093	. . . . . 6 wff y ~<_ x
21	11	csn 3673	. . . . . . 7 class { om }
22	11, 21	cun 3199	. . . . . 6 class ( om u. { om } )
23	20, 17, 22	crab 2515	. . . . 5 class { y e. ( om u. { om } ) | y ~<_ x }
24	23	cuni 3898	. . . 4 class U. { y e. ( om u. { om } ) | y ~<_ x }
25	2, 16, 24	cmpt 4155	. . 3 class ( x e. _V |-> U. { y e. ( om u. { om } ) | y ~<_ x } )
26	15, 25	ccom 4735	. 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  11086  hashennn  11088