Definition df-eldisj 35974

Description: Define the disjoint elementhood 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 35996.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 36042 with dfeldisj5 35988. See also the comments of ~? dfmembpart2 and of ~? df-parts . (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 35523	. 2 wff ElDisj 𝐴
3		cep 5457	. . . . 5 class E
4	3	ccnv 5547	. . . 4 class ◡ E
5	4, 1	cres 5550	. . 3 class (◡ E ↾ 𝐴)
6	5	wdisjALTV 35521	. 2 wff Disj (◡ E ↾ 𝐴)
7	2, 6	wb 208	1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))

This definition is referenced by:  dfeldisj2  35985  dfeldisj3  35986  dfeldisj4  35987  eleldisjseldisj  35996  eldisjss  36005  eldisjeq  36008