Theorem enq0ex 7649

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 4689	. . . 4 ⊢ ω ∈ V
2		niex 7522	. . . 4 ⊢ N ∈ V
3	1, 2	xpex 4840	. . 3 ⊢ (ω × N) ∈ V
4	3, 3	xpex 4840	. 2 ⊢ ((ω × N) × (ω × N)) ∈ V
5		df-enq0 7634	. . 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 4798	. . 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 3257	. 2 ⊢ ~Q0 ⊆ ((ω × N) × (ω × N))
8	4, 7	ssexi 4225	1 ⊢ ~Q0 ∈ V

Syntax hints:   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2800  ⟨cop 3670  {copab 4147  ωcom 4686   × cxp 4721  (class class class)co 6013   ·_o comu 6575  Ncnpi 7482   ~_Q0 ceq0 7496

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-opab 4149  df-iom 4687  df-xp 4729  df-ni 7514  df-enq0 7634

This theorem is referenced by:  nqnq0  7651  addnnnq0  7659  mulnnnq0  7660  addclnq0  7661  mulclnq0  7662  prarloclemcalc  7712