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Metamath Proof Explorer

Definition df-en

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

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

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cen	⊢ ≈
1		vx	⊢ 𝑥
2		vy	⊢ 𝑦
3		vf	⊢ 𝑓
4	3	cv	⊢ 𝑓
5	1	cv	⊢ 𝑥
6	2	cv	⊢ 𝑦
7	5 6 4	wf1o	⊢ 𝑓 : 𝑥 –1-1-onto→ 𝑦
8	7 3	wex	⊢ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑦
9	8 1 2	copab	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑦 }
10	0 9	wceq	⊢ ≈ = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑦 }