Theorem enqex 7733

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 7685	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4893	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4893	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7720	. . 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 4851	. . 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 3280	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4273	1 ⊢ ~Q ∈ V

This proof depends on syntax axioms:   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  ⟨cop 3712  {copab 4192   × cxp 4774  (class class class)co 6088  Ncnpi 7645   ·_N cmi 7647   ~_Q ceq 7652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-opab 4194  df-iom 4740  df-xp 4782  df-ni 7677  df-enq 7720

This theorem is used by:  1nq  7739  addpipqqs  7743  mulpipqqs  7746  ordpipqqs  7747  addclnq  7748  mulclnq  7749  dmaddpq  7752  dmmulpq  7753  recexnq  7763  ltexnqq  7781  prarloclemarch  7791  prarloclemarch2  7792  nnnq  7795  nqpnq0nq  7826  prarloclemlt  7866  prarloclemlo  7867  prarloclemcalc  7875  nqprm  7915