Definition df-eldisj 37880

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 37902.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 38045 with dfeldisj5 37894. See also the comments of dfmembpart2 37943 and of df-parts 37938. (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 37382	. 2 wff ElDisj 𝐴
3		cep 5578	. . . . 5 class E
4	3	ccnv 5674	. . . 4 class ◡ E
5	4, 1	cres 5677	. . 3 class (◡ E ↾ 𝐴)
6	5	wdisjALTV 37380	. 2 wff Disj (◡ E ↾ 𝐴)
7	2, 6	wb 205	1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))

This definition is referenced by:  dfeldisj2  37891  dfeldisj3  37892  dfeldisj4  37893  eleldisjseldisj  37902  eldisjss  37911  eldisjeq  37914  eldisjn0elb  37918  dfmembpart2  37943  eldisjim  37957  eldisjim2  37958  eldisjn0el  37979  eldisjlem19  37983  eqvreldisj3  37999