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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 37537.
As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 37680 with dfeldisj5 37529. See also the comments of dfmembpart2 37578 and of df-parts 37573. (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 37017 | . 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 37015 | . 2 wff Disj (◡ E ↾ 𝐴) |
7 | 2, 6 | wb 205 | 1 wff ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) |
Colors of variables: wff setvar class |
This definition is referenced by: dfeldisj2 37526 dfeldisj3 37527 dfeldisj4 37528 eleldisjseldisj 37537 eldisjss 37546 eldisjeq 37549 eldisjn0elb 37553 dfmembpart2 37578 eldisjim 37592 eldisjim2 37593 eldisjn0el 37614 eldisjlem19 37618 eqvreldisj3 37634 |
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