Definition df-en 8669

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 8678. (Contributed by NM, 28-Mar-1998.)

Assertion

Ref	Expression
df-en	⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}

Distinct variable group:   𝑥,𝑦,𝑓

Detailed syntax breakdown of Definition df-en

Step	Hyp	Ref	Expression
1		cen 8665	. 2 class ≈
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1542	. . . . 5 class 𝑥
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1542	. . . . 5 class 𝑦
6		vf	. . . . . 6 setvar 𝑓
7	6	cv 1542	. . . . 5 class 𝑓
8	3, 5, 7	wf1o 6414	. . . 4 wff 𝑓:𝑥–1-1-onto→𝑦
9	8, 6	wex 1787	. . 3 wff ∃𝑓 𝑓:𝑥–1-1-onto→𝑦
10	9, 2, 4	copab 5132	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
11	1, 10	wceq 1543	1 wff ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}

This definition is referenced by:  relen  8673  breng  8677  brenOLD  8679  enssdom  8697