Definition df-eldisj 38725

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 38747.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 38890 with dfeldisj5 38739. See also the comments of dfmembpart2 38788 and of df-parts 38783. (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 38235	. 2 wff ElDisj 𝐴
3		cep 5552	. . . . 5 class E
4	3	ccnv 5653	. . . 4 class ◡ E
5	4, 1	cres 5656	. . 3 class (◡ E ↾ 𝐴)
6	5	wdisjALTV 38233	. 2 wff Disj (◡ E ↾ 𝐴)
7	2, 6	wb 206	1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))

This definition is referenced by:  dfeldisj2  38736  dfeldisj3  38737  dfeldisj4  38738  eleldisjseldisj  38747  eldisjss  38756  eldisjeq  38759  eldisjn0elb  38763  dfmembpart2  38788  eldisjim  38802  eldisjim2  38803  eldisjn0el  38824  eldisjlem19  38828  eqvreldisj3  38844