Theorem enqex 7570

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 7522	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4840	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4840	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7557	. . 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 4798	. . 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 3257	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4225	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2800  ⟨cop 3670  {copab 4147   × cxp 4721  (class class class)co 6013  Ncnpi 7482   ·_N cmi 7484   ~_Q ceq 7489

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-opab 4149  df-iom 4687  df-xp 4729  df-ni 7514  df-enq 7557

This theorem is referenced by:  1nq  7576  addpipqqs  7580  mulpipqqs  7583  ordpipqqs  7584  addclnq  7585  mulclnq  7586  dmaddpq  7589  dmmulpq  7590  recexnq  7600  ltexnqq  7618  prarloclemarch  7628  prarloclemarch2  7629  nnnq  7632  nqpnq0nq  7663  prarloclemlt  7703  prarloclemlo  7704  prarloclemcalc  7712  nqprm  7752