Definition df-eldisj 38815

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 38837.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 38981 with dfeldisj5 38829. See also the comments of dfmembpart2 38878 and of df-parts 38873. (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 38268	. 2 wff ElDisj 𝐴
3		cep 5513	. . . . 5 class E
4	3	ccnv 5613	. . . 4 class ◡ E
5	4, 1	cres 5616	. . 3 class (◡ E ↾ 𝐴)
6	5	wdisjALTV 38266	. 2 wff Disj (◡ E ↾ 𝐴)
7	2, 6	wb 206	1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))

This definition is referenced by:  dfeldisj2  38826  dfeldisj3  38827  dfeldisj4  38828  eleldisjseldisj  38837  eldisjss  38846  eldisjeq  38849  eldisjn0elb  38853  dfmembpart2  38878  eldisjim  38892  eldisjim2  38893  eldisjn0el  38914  eldisjlem19  38918  eqvreldisj3  38934