Theorem enqex 7579

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 7531	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4842	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4842	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7566	. . 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 4800	. . 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 3259	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4227	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802  ⟨cop 3672  {copab 4149   × cxp 4723  (class class class)co 6017  Ncnpi 7491   ·_N cmi 7493   ~_Q ceq 7498

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-opab 4151  df-iom 4689  df-xp 4731  df-ni 7523  df-enq 7566

This theorem is referenced by:  1nq  7585  addpipqqs  7589  mulpipqqs  7592  ordpipqqs  7593  addclnq  7594  mulclnq  7595  dmaddpq  7598  dmmulpq  7599  recexnq  7609  ltexnqq  7627  prarloclemarch  7637  prarloclemarch2  7638  nnnq  7641  nqpnq0nq  7672  prarloclemlt  7712  prarloclemlo  7713  prarloclemcalc  7721  nqprm  7761