Theorem enqex 7444

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 7396	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4779	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4779	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7431	. . 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 4738	. . 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 3216	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4172	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763  ⟨cop 3626  {copab 4094   × cxp 4662  (class class class)co 5925  Ncnpi 7356   ·_N cmi 7358   ~_Q ceq 7363

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-opab 4096  df-iom 4628  df-xp 4670  df-ni 7388  df-enq 7431

This theorem is referenced by:  1nq  7450  addpipqqs  7454  mulpipqqs  7457  ordpipqqs  7458  addclnq  7459  mulclnq  7460  dmaddpq  7463  dmmulpq  7464  recexnq  7474  ltexnqq  7492  prarloclemarch  7502  prarloclemarch2  7503  nnnq  7506  nqpnq0nq  7537  prarloclemlt  7577  prarloclemlo  7578  prarloclemcalc  7586  nqprm  7626