Theorem elpqn 10839

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 10826	. . 3 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2	1	ssrab3 4023	. 2 ⊢ Q ⊆ (N × N)
3	2	sseli 3918	1 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086   × cxp 5622  ‘cfv 6492  2^nd c2nd 7934  Ncnpi 10758   <_N clti 10761   ~_Q ceq 10765  Qcnq 10766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-ss 3907  df-nq 10826

This theorem is referenced by:  nqereu  10843  nqerid  10847  enqeq  10848  addpqnq  10852  mulpqnq  10855  ordpinq  10857  addclnq  10859  mulclnq  10861  addnqf  10862  mulnqf  10863  adderpq  10870  mulerpq  10871  addassnq  10872  mulassnq  10873  distrnq  10875  mulidnq  10877  recmulnq  10878  ltsonq  10883  lterpq  10884  ltanq  10885  ltmnq  10886  ltexnq  10889  archnq  10894  wuncn  11084