Theorem elpqn 9707

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 9694	. . 3 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2		ssrab2 3672	. . 3 ⊢ {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} ⊆ (N × N)
3	1, 2	eqsstri 3620	. 2 ⊢ Q ⊆ (N × N)
4	3	sseli 3584	1 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1987  ∀wral 2908  {crab 2912   class class class wbr 4623   × cxp 5082  ‘cfv 5857  2^nd c2nd 7127  Ncnpi 9626   <_N clti 9629   ~_Q ceq 9633  Qcnq 9634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-in 3567  df-ss 3574  df-nq 9694

This theorem is referenced by:  nqereu  9711  nqerid  9715  enqeq  9716  addpqnq  9720  mulpqnq  9723  ordpinq  9725  addclnq  9727  mulclnq  9729  addnqf  9730  mulnqf  9731  adderpq  9738  mulerpq  9739  addassnq  9740  mulassnq  9741  distrnq  9743  mulidnq  9745  recmulnq  9746  ltsonq  9751  lterpq  9752  ltanq  9753  ltmnq  9754  ltexnq  9757  archnq  9762  wuncn  9951