| Mathbox for Peter Mazsa |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > df-eldisj | Structured version Visualization version GIF version | ||
| 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 38730.
As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 38873 with dfeldisj5 38722. See also the comments of dfmembpart2 38771 and of df-parts 38766. (Contributed by Peter Mazsa, 17-Jul-2021.) |
| Ref | Expression |
|---|---|
| df-eldisj | ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cA | . . 3 class 𝐴 | |
| 2 | 1 | weldisj 38218 | . 2 wff ElDisj 𝐴 |
| 3 | cep 5583 | . . . . 5 class E | |
| 4 | 3 | ccnv 5684 | . . . 4 class ◡ E |
| 5 | 4, 1 | cres 5687 | . . 3 class (◡ E ↾ 𝐴) |
| 6 | 5 | wdisjALTV 38216 | . 2 wff Disj (◡ E ↾ 𝐴) |
| 7 | 2, 6 | wb 206 | 1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) |
| Colors of variables: wff setvar class |
| This definition is referenced by: dfeldisj2 38719 dfeldisj3 38720 dfeldisj4 38721 eleldisjseldisj 38730 eldisjss 38739 eldisjeq 38742 eldisjn0elb 38746 dfmembpart2 38771 eldisjim 38785 eldisjim2 38786 eldisjn0el 38807 eldisjlem19 38811 eqvreldisj3 38827 |
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