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Theorem prpssnq 10389
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 10387 . 2 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
2 simpl3 1190 . 2 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → 𝐴Q)
31, 2sylbi 220 1 (𝐴P𝐴Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wal 1536  wcel 2115  wral 3126  wrex 3127  Vcvv 3471  wpss 3911  c0 4266   class class class wbr 5039  Qcnq 10251   <Q cltq 10257  Pcnp 10258
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-v 3473  df-in 3917  df-ss 3927  df-pss 3929  df-np 10380
This theorem is referenced by:  elprnq  10390  npomex  10395  genpnnp  10404  prlem934  10432  ltexprlem2  10436  reclem2pr  10447  suplem1pr  10451  wuncn  10569
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