Theorem enq0ex 7148

Description: The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)

Assertion

Ref	Expression
enq0ex	⊢ ~Q0 ∈ V

Proof of Theorem enq0ex

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omex 4445	. . . 4 ⊢ ω ∈ V
2		niex 7021	. . . 4 ⊢ N ∈ V
3	1, 2	xpex 4592	. . 3 ⊢ (ω × N) ∈ V
4	3, 3	xpex 4592	. 2 ⊢ ((ω × N) × (ω × N)) ∈ V
5		df-enq0 7133	. . 3 ⊢ ~Q0 = {⟨𝑣, 𝑢⟩ ∣ ((𝑣 ∈ (ω × N) ∧ 𝑢 ∈ (ω × N)) ∧ ∃𝑥∃𝑦∃𝑧∃𝑤((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝑢 = ⟨𝑧, 𝑤⟩) ∧ (𝑥 ·o 𝑤) = (𝑦 ·o 𝑧)))}
6		opabssxp 4551	. . 3 ⊢ {⟨𝑣, 𝑢⟩ ∣ ((𝑣 ∈ (ω × N) ∧ 𝑢 ∈ (ω × N)) ∧ ∃𝑥∃𝑦∃𝑧∃𝑤((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝑢 = ⟨𝑧, 𝑤⟩) ∧ (𝑥 ·o 𝑤) = (𝑦 ·o 𝑧)))} ⊆ ((ω × N) × (ω × N))
7	5, 6	eqsstri 3079	. 2 ⊢ ~Q0 ⊆ ((ω × N) × (ω × N))
8	4, 7	ssexi 4006	1 ⊢ ~Q0 ∈ V

Syntax hints:   ∧ wa 103   = wceq 1299  ∃wex 1436   ∈ wcel 1448  Vcvv 2641  ⟨cop 3477  {copab 3928  ωcom 4442   × cxp 4475  (class class class)co 5706   ·_o comu 6241  Ncnpi 6981   ~_Q0 ceq0 6995

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-opab 3930  df-iom 4443  df-xp 4483  df-ni 7013  df-enq0 7133

This theorem is referenced by:  nqnq0  7150  addnnnq0  7158  mulnnnq0  7159  addclnq0  7160  mulclnq0  7161  prarloclemcalc  7211