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Metamath Proof Explorer

Theorem enrex

Description: The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995) (New usage is discouraged.)

Ref Expression
Assertion enrex ~R ∈ V

Proof

Step Hyp Ref Expression
1 npex ⊢ P ∈ V
2 1 1 xpex ⊢ ( P × P ) ∈ V
3 2 2 xpex ⊢ ( ( P × P ) × ( P × P ) ) ∈ V
4 df-enr ⊢ ~R = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ ( P × P ) ∧ 𝑦 ∈ ( P × P ) ) ∧ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) ) ) }
5 opabssxp ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ ( P × P ) ∧ 𝑦 ∈ ( P × P ) ) ∧ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) ) ) } ⊆ ( ( P × P ) × ( P × P ) )
6 4 5 eqsstri ⊢ ~R ⊆ ( ( P × P ) × ( P × P ) )
7 3 6 ssexi ⊢ ~R ∈ V