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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 8317. (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 8305 | . 2 class ≈ | |
2 | vx | . . . . . 6 setvar 𝑥 | |
3 | 2 | cv 1506 | . . . . 5 class 𝑥 |
4 | vy | . . . . . 6 setvar 𝑦 | |
5 | 4 | cv 1506 | . . . . 5 class 𝑦 |
6 | vf | . . . . . 6 setvar 𝑓 | |
7 | 6 | cv 1506 | . . . . 5 class 𝑓 |
8 | 3, 5, 7 | wf1o 6189 | . . . 4 wff 𝑓:𝑥–1-1-onto→𝑦 |
9 | 8, 6 | wex 1742 | . . 3 wff ∃𝑓 𝑓:𝑥–1-1-onto→𝑦 |
10 | 9, 2, 4 | copab 4992 | . 2 class {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
11 | 1, 10 | wceq 1507 | 1 wff ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
Colors of variables: wff setvar class |
This definition is referenced by: relen 8313 bren 8317 enssdom 8333 |
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