Theorem enq0ex 7770

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 4720	. . . 4 ⊢ ω ∈ V
2		niex 7643	. . . 4 ⊢ N ∈ V
3	1, 2	xpex 4871	. . 3 ⊢ (ω × N) ∈ V
4	3, 3	xpex 4871	. 2 ⊢ ((ω × N) × (ω × N)) ∈ V
5		df-enq0 7755	. . 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 4829	. . 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 3274	. 2 ⊢ ~Q0 ⊆ ((ω × N) × (ω × N))
8	4, 7	ssexi 4253	1 ⊢ ~Q0 ∈ V

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2205  Vcvv 2815  ⟨cop 3697  {copab 4175  ωcom 4717   × cxp 4752  (class class class)co 6058   ·_o comu 6658  Ncnpi 7603   ~_Q0 ceq0 7617

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-opab 4177  df-iom 4718  df-xp 4760  df-ni 7635  df-enq0 7755

This theorem is referenced by:  nqnq0  7772  addnnnq0  7780  mulnnnq0  7781  addclnq0  7782  mulclnq0  7783  prarloclemcalc  7833