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Metamath Proof Explorer

Theorem enqex

Description: The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995) (New usage is discouraged.)

Ref Expression
Assertion enqex ~Q ∈ V

Proof

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