Definition df-eldisj 37881

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 37903.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 38046 with dfeldisj5 37895. See also the comments of dfmembpart2 37944 and of df-parts 37939. (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 37383	. 2 wff ElDisj 𝐴
3		cep 5579	. . . . 5 class E
4	3	ccnv 5675	. . . 4 class ◡ E
5	4, 1	cres 5678	. . 3 class (◡ E ↾ 𝐴)
6	5	wdisjALTV 37381	. 2 wff Disj (◡ E ↾ 𝐴)
7	2, 6	wb 205	1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))

This definition is referenced by:  dfeldisj2  37892  dfeldisj3  37893  dfeldisj4  37894  eleldisjseldisj  37903  eldisjss  37912  eldisjeq  37915  eldisjn0elb  37919  dfmembpart2  37944  eldisjim  37958  eldisjim2  37959  eldisjn0el  37980  eldisjlem19  37984  eqvreldisj3  38000