Definition df-eldisj 38663

Description: Define the disjoint element relation predicate, i.e., the disjoint elementhood predicate. Read: the elements of 𝐴 are disjoint. The element of the disjoint elements class and the disjoint elementhood predicate are the same, that is (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴) when 𝐴 is a set, see eleldisjseldisj 38685.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 38828 with dfeldisj5 38677. See also the comments of dfmembpart2 38726 and of df-parts 38721. (Contributed by Peter Mazsa, 17-Jul-2021.)

Assertion

Ref	Expression
df-eldisj	⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))

Detailed syntax breakdown of Definition df-eldisj

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	weldisj 38171	. 2 wff ElDisj 𝐴
3		cep 5598	. . . . 5 class E
4	3	ccnv 5699	. . . 4 class ◡ E
5	4, 1	cres 5702	. . 3 class (◡ E ↾ 𝐴)
6	5	wdisjALTV 38169	. 2 wff Disj (◡ E ↾ 𝐴)
7	2, 6	wb 206	1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))

This definition is referenced by:  dfeldisj2  38674  dfeldisj3  38675  dfeldisj4  38676  eleldisjseldisj  38685  eldisjss  38694  eldisjeq  38697  eldisjn0elb  38701  dfmembpart2  38726  eldisjim  38740  eldisjim2  38741  eldisjn0el  38762  eldisjlem19  38766  eqvreldisj3  38782