Theorem elpqn 9626

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	⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))

Proof of Theorem elpqn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nq 9613	. . 3 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2		ssrab2 3650	. . 3 ⊢ {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} ⊆ (N × N)
3	1, 2	eqsstri 3598	. 2 ⊢ Q ⊆ (N × N)
4	3	sseli 3564	1 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1977  ∀wral 2896  {crab 2900   class class class wbr 4583   × cxp 5036  ‘cfv 5804  2^nd c2nd 7058  Ncnpi 9545   <_N clti 9548   ~_Q ceq 9552  Qcnq 9553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-in 3547  df-ss 3554  df-nq 9613

This theorem is referenced by:  nqereu  9630  nqerid  9634  enqeq  9635  addpqnq  9639  mulpqnq  9642  ordpinq  9644  addclnq  9646  mulclnq  9648  addnqf  9649  mulnqf  9650  adderpq  9657  mulerpq  9658  addassnq  9659  mulassnq  9660  distrnq  9662  mulidnq  9664  recmulnq  9665  ltsonq  9670  lterpq  9671  ltanq  9672  ltmnq  9673  ltexnq  9676  archnq  9681  wuncn  9870