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