Theorem enqex 7301

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 7253	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4719	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4719	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7288	. . 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 4678	. . 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 3174	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4120	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726  ⟨cop 3579  {copab 4042   × cxp 4602  (class class class)co 5842  Ncnpi 7213   ·_N cmi 7215   ~_Q ceq 7220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-opab 4044  df-iom 4568  df-xp 4610  df-ni 7245  df-enq 7288

This theorem is referenced by:  1nq  7307  addpipqqs  7311  mulpipqqs  7314  ordpipqqs  7315  addclnq  7316  mulclnq  7317  dmaddpq  7320  dmmulpq  7321  recexnq  7331  ltexnqq  7349  prarloclemarch  7359  prarloclemarch2  7360  nnnq  7363  nqpnq0nq  7394  prarloclemlt  7434  prarloclemlo  7435  prarloclemcalc  7443  nqprm  7483