Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-disj | Structured version Visualization version GIF version |
Description: A collection of classes 𝐵(𝑥) is disjoint when for each element 𝑦, it is in 𝐵(𝑥) for at most one 𝑥. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
df-disj | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . 3 setvar 𝑥 | |
2 | cA | . . 3 class 𝐴 | |
3 | cB | . . 3 class 𝐵 | |
4 | 1, 2, 3 | wdisj 5040 | . 2 wff Disj 𝑥 ∈ 𝐴 𝐵 |
5 | vy | . . . . . 6 setvar 𝑦 | |
6 | 5 | cv 1538 | . . . . 5 class 𝑦 |
7 | 6, 3 | wcel 2107 | . . . 4 wff 𝑦 ∈ 𝐵 |
8 | 7, 1, 2 | wrmo 3068 | . . 3 wff ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
9 | 8, 5 | wal 1537 | . 2 wff ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
10 | 4, 9 | wb 205 | 1 wff (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
Colors of variables: wff setvar class |
This definition is referenced by: dfdisj2 5042 disjss2 5043 cbvdisj 5050 nfdisj1 5054 disjor 5055 disjiun 5062 cbvdisjf 30919 disjss1f 30920 disjxun0 30922 disjorf 30927 disjin 30934 disjin2 30935 disjrdx 30939 ddemeas 32213 iccpartdisj 44900 |
Copyright terms: Public domain | W3C validator |