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Theorem prpssnq 9691
Description: A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prpssnq (𝐴P𝐴Q)

Proof of Theorem prpssnq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnpi 9689 . 2 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
2 simpl3 1059 . 2 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → 𝐴Q)
31, 2sylbi 206 1 (𝐴P𝐴Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031  wal 1473  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  wpss 3541  c0 3874   class class class wbr 4583  Qcnq 9553   <Q cltq 9559  Pcnp 9560
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-3an 1033  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-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-pss 3556  df-np 9682
This theorem is referenced by:  elprnq  9692  npomex  9697  genpnnp  9706  prlem934  9734  ltexprlem2  9738  reclem2pr  9749  suplem1pr  9753  wuncn  9870
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