Definition df-eldisj 39543

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 39580.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 39748 with dfeldisj5 39564. See also the comments of dfmembpart2 39624 and of df-parts 39619. (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 38972	. 2 wff ElDisj 𝐴
3		cep 5554	. . . . 5 class E
4	3	ccnv 5654	. . . 4 class ◡ E
5	4, 1	cres 5657	. . 3 class (◡ E ↾ 𝐴)
6	5	wdisjALTV 38970	. 2 wff Disj (◡ E ↾ 𝐴)
7	2, 6	wb 209	1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))

This definition is used by:  dfeldisj2  39561  dfeldisj3  39562  dfeldisj4  39563  eleldisjseldisj  39580  eldisjss  39589  eldisjeq  39592  eldisjn0elb  39596  dfmembpart2  39624  eldisjim  39638  eldisjim2  39639  eldisjn0el  39660  eldisjlem19  39664  eqvreldisj3  39680