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Theorem prpssnq 10099
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 10097 . 2 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
2 simpl3 1247 . 2 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → 𝐴Q)
31, 2sylbi 209 1 (𝐴P𝐴Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108  wal 1651  wcel 2157  wral 3088  wrex 3089  Vcvv 3384  wpss 3769  c0 4114   class class class wbr 4842  Qcnq 9961   <Q cltq 9967  Pcnp 9968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-v 3386  df-in 3775  df-ss 3782  df-pss 3784  df-np 10090
This theorem is referenced by:  elprnq  10100  npomex  10105  genpnnp  10114  prlem934  10142  ltexprlem2  10146  reclem2pr  10157  suplem1pr  10161  wuncn  10278
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