Theorem enqex 7727

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 7679	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4891	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4891	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7714	. . 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 4849	. . 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 4271	1 ⊢ ~Q ∈ V

This proof depends on syntax axioms:   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  ⟨cop 3712  {copab 4191   × cxp 4772  (class class class)co 6085  Ncnpi 7639   ·_N cmi 7641   ~_Q ceq 7646

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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

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 4193  df-iom 4738  df-xp 4780  df-ni 7671  df-enq 7714

This theorem is used by:  1nq  7733  addpipqqs  7737  mulpipqqs  7740  ordpipqqs  7741  addclnq  7742  mulclnq  7743  dmaddpq  7746  dmmulpq  7747  recexnq  7757  ltexnqq  7775  prarloclemarch  7785  prarloclemarch2  7786  nnnq  7789  nqpnq0nq  7820  prarloclemlt  7860  prarloclemlo  7861  prarloclemcalc  7869  nqprm  7909