Definition df-eldisj 39291

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 39328.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 39496 with dfeldisj5 39312. See also the comments of dfmembpart2 39372 and of df-parts 39367. (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 38720	. 2 wff ElDisj 𝐴
3		cep 5546	. . . . 5 class E
4	3	ccnv 5646	. . . 4 class ◡ E
5	4, 1	cres 5649	. . 3 class (◡ E ↾ 𝐴)
6	5	wdisjALTV 38718	. 2 wff Disj (◡ E ↾ 𝐴)
7	2, 6	wb 208	1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))

This definition is referenced by:  dfeldisj2  39309  dfeldisj3  39310  dfeldisj4  39311  eleldisjseldisj  39328  eldisjss  39337  eldisjeq  39340  eldisjn0elb  39344  dfmembpart2  39372  eldisjim  39386  eldisjim2  39387  eldisjn0el  39408  eldisjlem19  39412  eqvreldisj3  39428