Theorem enqex 7388

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 7340	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4759	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4759	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7375	. . 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 4718	. . 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 3202	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4156	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2160  Vcvv 2752  ⟨cop 3610  {copab 4078   × cxp 4642  (class class class)co 5895  Ncnpi 7300   ·_N cmi 7302   ~_Q ceq 7307

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-opab 4080  df-iom 4608  df-xp 4650  df-ni 7332  df-enq 7375

This theorem is referenced by:  1nq  7394  addpipqqs  7398  mulpipqqs  7401  ordpipqqs  7402  addclnq  7403  mulclnq  7404  dmaddpq  7407  dmmulpq  7408  recexnq  7418  ltexnqq  7436  prarloclemarch  7446  prarloclemarch2  7447  nnnq  7450  nqpnq0nq  7481  prarloclemlt  7521  prarloclemlo  7522  prarloclemcalc  7530  nqprm  7570