Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-eprel | Structured version Visualization version GIF version |
Description: Define the membership relation (also called "epsilon relation" since it is sometimes denoted by the lowercase Greek letter "epsilon"). Similar to Definition 6.22 of [TakeutiZaring] p. 30. The membership relation and the membership predicate agree, that is, (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵), when 𝐵 is a set (see epelg 5492). Thus, ⊢ 5 E {1, 5} (ex-eprel 28783). (Contributed by NM, 13-Aug-1995.) |
Ref | Expression |
---|---|
df-eprel | ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cep 5490 | . 2 class E | |
2 | vx | . . . 4 setvar 𝑥 | |
3 | vy | . . . 4 setvar 𝑦 | |
4 | 2, 3 | wel 2107 | . . 3 wff 𝑥 ∈ 𝑦 |
5 | 4, 2, 3 | copab 5136 | . 2 class {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} |
6 | 1, 5 | wceq 1539 | 1 wff E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} |
Colors of variables: wff setvar class |
This definition is referenced by: epelg 5492 rele 5731 epinid0 9347 cnvepnep 9354 bj-epelg 35225 dfnelbr2 44721 |
Copyright terms: Public domain | W3C validator |