Theorem elpqn 10880

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 10867	. . 3 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2	1	ssrab3 4035	. 2 ⊢ Q ⊆ (N × N)
3	2	sseli 3932	1 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2141  ∀wral 3075   class class class wbr 5099   × cxp 5643  ‘cfv 6517  2^nd c2nd 7965  Ncnpi 10799   <_N clti 10802   ~_Q ceq 10806  Qcnq 10807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-ss 3921  df-nq 10867

This theorem is referenced by:  nqereu  10884  nqerid  10888  enqeq  10889  addpqnq  10893  mulpqnq  10896  ordpinq  10898  addclnq  10900  mulclnq  10902  addnqf  10903  mulnqf  10904  adderpq  10911  mulerpq  10912  addassnq  10913  mulassnq  10914  distrnq  10916  mulidnq  10918  recmulnq  10919  ltsonq  10924  lterpq  10925  ltanq  10926  ltmnq  10927  ltexnq  10930  archnq  10935  wuncn  11125