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Metamath Proof Explorer

Theorem ideqg

Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011)

Ref Expression
Assertion ideqg ( 𝐵 ∈ 𝑉 → ( 𝐴 I 𝐵 ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 id ⊢ ( 𝐵 ∈ 𝑉 → 𝐵 ∈ 𝑉 )
2 reli ⊢ Rel I
3 2 brrelex1i ⊢ ( 𝐴 I 𝐵 → 𝐴 ∈ V )
4 1 3 anim12ci ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 I 𝐵 ) → ( 𝐴 ∈ V ∧ 𝐵 ∈ 𝑉 ) )
5 eleq1 ⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉 ) )
6 5 biimparc ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ 𝑉 )
7 6 elexd ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ V )
8 simpl ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → 𝐵 ∈ 𝑉 )
9 7 8 jca ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → ( 𝐴 ∈ V ∧ 𝐵 ∈ 𝑉 ) )
10 eqeq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 = 𝑦 ↔ 𝐴 = 𝑦 ) )
11 eqeq2 ⊢ ( 𝑦 = 𝐵 → ( 𝐴 = 𝑦 ↔ 𝐴 = 𝐵 ) )
12 df-id ⊢ I = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 = 𝑦 }
13 10 11 12 brabg ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 I 𝐵 ↔ 𝐴 = 𝐵 ) )
14 4 9 13 pm5.21nd ⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 I 𝐵 ↔ 𝐴 = 𝐵 ) )