Theorem enqex 7623

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 7575	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4848	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4848	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7610	. . 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 4806	. . 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 3260	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4232	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803  ⟨cop 3676  {copab 4154   × cxp 4729  (class class class)co 6028  Ncnpi 7535   ·_N cmi 7537   ~_Q ceq 7542

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-enq 7610

This theorem is referenced by:  1nq  7629  addpipqqs  7633  mulpipqqs  7636  ordpipqqs  7637  addclnq  7638  mulclnq  7639  dmaddpq  7642  dmmulpq  7643  recexnq  7653  ltexnqq  7671  prarloclemarch  7681  prarloclemarch2  7682  nnnq  7685  nqpnq0nq  7716  prarloclemlt  7756  prarloclemlo  7757  prarloclemcalc  7765  nqprm  7805