Theorem enqex 7116

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 7068	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4614	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4614	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7103	. . 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 4573	. . 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 3095	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4026	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 103   = wceq 1314  ∃wex 1451   ∈ wcel 1463  Vcvv 2657  ⟨cop 3496  {copab 3948   × cxp 4497  (class class class)co 5728  Ncnpi 7028   ·_N cmi 7030   ~_Q ceq 7035

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-iinf 4462

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-opab 3950  df-iom 4465  df-xp 4505  df-ni 7060  df-enq 7103

This theorem is referenced by:  1nq  7122  addpipqqs  7126  mulpipqqs  7129  ordpipqqs  7130  addclnq  7131  mulclnq  7132  dmaddpq  7135  dmmulpq  7136  recexnq  7146  ltexnqq  7164  prarloclemarch  7174  prarloclemarch2  7175  nnnq  7178  nqpnq0nq  7209  prarloclemlt  7249  prarloclemlo  7250  prarloclemcalc  7258  nqprm  7298