Definition df-eldisj 36100

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 36122.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 36168 with dfeldisj5 36114. 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 35649	. 2 wff ElDisj 𝐴
3		cep 5429	. . . . 5 class E
4	3	ccnv 5518	. . . 4 class ◡ E
5	4, 1	cres 5521	. . 3 class (◡ E ↾ 𝐴)
6	5	wdisjALTV 35647	. 2 wff Disj (◡ E ↾ 𝐴)
7	2, 6	wb 209	1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))

This definition is referenced by:  dfeldisj2  36111  dfeldisj3  36112  dfeldisj4  36113  eleldisjseldisj  36122  eldisjss  36131  eldisjeq  36134