Theorem idssen

Description: Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998) (Revised by Mario Carneiro, 15-Nov-2014)

	Ref	Expression
Assertion	idssen	⊢ I ⊆ ≈

Proof

Step	Hyp	Ref	Expression
1		reli	⊢ Rel I
2		vex	⊢ 𝑦 ∈ V
3	2	ideq	⊢ ( 𝑥 I 𝑦 ↔ 𝑥 = 𝑦 )
4		eqeng	⊢ ( 𝑥 ∈ V → ( 𝑥 = 𝑦 → 𝑥 ≈ 𝑦 ) )
5	4	elv	⊢ ( 𝑥 = 𝑦 → 𝑥 ≈ 𝑦 )
6	3 5	sylbi	⊢ ( 𝑥 I 𝑦 → 𝑥 ≈ 𝑦 )
7		df-br	⊢ ( 𝑥 I 𝑦 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ I )
8		df-br	⊢ ( 𝑥 ≈ 𝑦 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ ≈ )
9	6 7 8	3imtr3i	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ I → ⟨ 𝑥 , 𝑦 ⟩ ∈ ≈ )
10	1 9	relssi	⊢ I ⊆ ≈