Theorem enqex 7358

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 7310	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4741	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4741	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7345	. . 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 4700	. . 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 3187	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4141	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 104   = wceq 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2737  ⟨cop 3595  {copab 4063   × cxp 4624  (class class class)co 5874  Ncnpi 7270   ·_N cmi 7272   ~_Q ceq 7277

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-opab 4065  df-iom 4590  df-xp 4632  df-ni 7302  df-enq 7345

This theorem is referenced by:  1nq  7364  addpipqqs  7368  mulpipqqs  7371  ordpipqqs  7372  addclnq  7373  mulclnq  7374  dmaddpq  7377  dmmulpq  7378  recexnq  7388  ltexnqq  7406  prarloclemarch  7416  prarloclemarch2  7417  nnnq  7420  nqpnq0nq  7451  prarloclemlt  7491  prarloclemlo  7492  prarloclemcalc  7500  nqprm  7540