Theorem enqex 7675

Description: The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)

Assertion

Ref	Expression
enqex	⊢ ~Q ∈ V

Proof of Theorem enqex

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

Step	Hyp	Ref	Expression
1		niex 7627	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4866	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4866	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7662	. . 3 ⊢ ~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
5		opabssxp 4824	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} ⊆ ((N × N) × (N × N))
6	4, 5	eqsstri 3270	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4248	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2813  ⟨cop 3692  {copab 4170   × cxp 4747  (class class class)co 6050  Ncnpi 7587   ·_N cmi 7589   ~_Q ceq 7594

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-opab 4172  df-iom 4713  df-xp 4755  df-ni 7619  df-enq 7662

This theorem is referenced by:  1nq  7681  addpipqqs  7685  mulpipqqs  7688  ordpipqqs  7689  addclnq  7690  mulclnq  7691  dmaddpq  7694  dmmulpq  7695  recexnq  7705  ltexnqq  7723  prarloclemarch  7733  prarloclemarch2  7734  nnnq  7737  nqpnq0nq  7768  prarloclemlt  7808  prarloclemlo  7809  prarloclemcalc  7817  nqprm  7857