Theorem enq0ex 7702

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 4697	. . . 4 ⊢ ω ∈ V
2		niex 7575	. . . 4 ⊢ N ∈ V
3	1, 2	xpex 4848	. . 3 ⊢ (ω × N) ∈ V
4	3, 3	xpex 4848	. 2 ⊢ ((ω × N) × (ω × N)) ∈ V
5		df-enq0 7687	. . 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 4806	. . 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 3260	. 2 ⊢ ~Q0 ⊆ ((ω × N) × (ω × N))
8	4, 7	ssexi 4232	1 ⊢ ~Q0 ∈ V

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803  ⟨cop 3676  {copab 4154  ωcom 4694   × cxp 4729  (class class class)co 6028   ·_o comu 6623  Ncnpi 7535   ~_Q0 ceq0 7549

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-opab 4156  df-iom 4695  df-xp 4737  df-ni 7567  df-enq0 7687

This theorem is referenced by:  nqnq0  7704  addnnnq0  7712  mulnnnq0  7713  addclnq0  7714  mulclnq0  7715  prarloclemcalc  7765