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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 8976. (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 8963 | . 2 class ≈ | |
2 | vx | . . . . . 6 setvar 𝑥 | |
3 | 2 | cv 1533 | . . . . 5 class 𝑥 |
4 | vy | . . . . . 6 setvar 𝑦 | |
5 | 4 | cv 1533 | . . . . 5 class 𝑦 |
6 | vf | . . . . . 6 setvar 𝑓 | |
7 | 6 | cv 1533 | . . . . 5 class 𝑓 |
8 | 3, 5, 7 | wf1o 6545 | . . . 4 wff 𝑓:𝑥–1-1-onto→𝑦 |
9 | 8, 6 | wex 1774 | . . 3 wff ∃𝑓 𝑓:𝑥–1-1-onto→𝑦 |
10 | 9, 2, 4 | copab 5207 | . 2 class {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
11 | 1, 10 | wceq 1534 | 1 wff ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
Colors of variables: wff setvar class |
This definition is referenced by: relen 8971 breng 8975 brenOLD 8977 enssdom 9000 |
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