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Theorem elpqn 9732
Description: Each positive fraction is an ordered pair of positive integers (the numerator and denominator, in "lowest terms". (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elpqn (𝐴Q𝐴 ∈ (N × N))

Proof of Theorem elpqn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nq 9719 . . 3 Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))}
2 ssrab2 3679 . . 3 {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))} ⊆ (N × N)
31, 2eqsstri 3627 . 2 Q ⊆ (N × N)
43sseli 3591 1 (𝐴Q𝐴 ∈ (N × N))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1988  wral 2909  {crab 2913   class class class wbr 4644   × cxp 5102  cfv 5876  2nd c2nd 7152  Ncnpi 9651   <N clti 9654   ~Q ceq 9658  Qcnq 9659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-in 3574  df-ss 3581  df-nq 9719
This theorem is referenced by:  nqereu  9736  nqerid  9740  enqeq  9741  addpqnq  9745  mulpqnq  9748  ordpinq  9750  addclnq  9752  mulclnq  9754  addnqf  9755  mulnqf  9756  adderpq  9763  mulerpq  9764  addassnq  9765  mulassnq  9766  distrnq  9768  mulidnq  9770  recmulnq  9771  ltsonq  9776  lterpq  9777  ltanq  9778  ltmnq  9779  ltexnq  9782  archnq  9787  wuncn  9976
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