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Mirrors > Home > MPE Home > Th. List > df-en | Structured version Visualization version GIF version |
Description: Define the equinumerosity relation. Definition of [Enderton] p. 129. We define ≈ to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren 8507. (Contributed by NM, 28-Mar-1998.) |
Ref | Expression |
---|---|
df-en | ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cen 8495 | . 2 class ≈ | |
2 | vx | . . . . . 6 setvar 𝑥 | |
3 | 2 | cv 1527 | . . . . 5 class 𝑥 |
4 | vy | . . . . . 6 setvar 𝑦 | |
5 | 4 | cv 1527 | . . . . 5 class 𝑦 |
6 | vf | . . . . . 6 setvar 𝑓 | |
7 | 6 | cv 1527 | . . . . 5 class 𝑓 |
8 | 3, 5, 7 | wf1o 6348 | . . . 4 wff 𝑓:𝑥–1-1-onto→𝑦 |
9 | 8, 6 | wex 1771 | . . 3 wff ∃𝑓 𝑓:𝑥–1-1-onto→𝑦 |
10 | 9, 2, 4 | copab 5120 | . 2 class {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
11 | 1, 10 | wceq 1528 | 1 wff ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
Colors of variables: wff setvar class |
This definition is referenced by: relen 8503 bren 8507 enssdom 8523 |
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