Theorem enq0ex 7469

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 4610	. . . 4 ⊢ ω ∈ V
2		niex 7342	. . . 4 ⊢ N ∈ V
3	1, 2	xpex 4759	. . 3 ⊢ (ω × N) ∈ V
4	3, 3	xpex 4759	. 2 ⊢ ((ω × N) × (ω × N)) ∈ V
5		df-enq0 7454	. . 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 4718	. . 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 3202	. 2 ⊢ ~Q0 ⊆ ((ω × N) × (ω × N))
8	4, 7	ssexi 4156	1 ⊢ ~Q0 ∈ V

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2160  Vcvv 2752  ⟨cop 3610  {copab 4078  ωcom 4607   × cxp 4642  (class class class)co 5897   ·_o comu 6440  Ncnpi 7302   ~_Q0 ceq0 7316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-opab 4080  df-iom 4608  df-xp 4650  df-ni 7334  df-enq0 7454

This theorem is referenced by:  nqnq0  7471  addnnnq0  7479  mulnnnq0  7480  addclnq0  7481  mulclnq0  7482  prarloclemcalc  7532