Theorem enqex 7309

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 7261	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4724	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4724	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7296	. . 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 4683	. . 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 3179	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4125	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730  ⟨cop 3584  {copab 4047   × cxp 4607  (class class class)co 5850  Ncnpi 7221   ·_N cmi 7223   ~_Q ceq 7228

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-opab 4049  df-iom 4573  df-xp 4615  df-ni 7253  df-enq 7296

This theorem is referenced by:  1nq  7315  addpipqqs  7319  mulpipqqs  7322  ordpipqqs  7323  addclnq  7324  mulclnq  7325  dmaddpq  7328  dmmulpq  7329  recexnq  7339  ltexnqq  7357  prarloclemarch  7367  prarloclemarch2  7368  nnnq  7371  nqpnq0nq  7402  prarloclemlt  7442  prarloclemlo  7443  prarloclemcalc  7451  nqprm  7491