MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prpssnq Structured version   Visualization version   GIF version

Theorem prpssnq 11031
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 11029 . 2 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
2 simpl3 1193 . 2 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → 𝐴Q)
31, 2sylbi 217 1 (𝐴P𝐴Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wal 1537  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  wpss 3951  c0 4332   class class class wbr 5142  Qcnq 10893   <Q cltq 10899  Pcnp 10900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3481  df-ss 3967  df-pss 3970  df-np 11022
This theorem is referenced by:  elprnq  11032  npomex  11037  genpnnp  11046  prlem934  11074  ltexprlem2  11078  reclem2pr  11089  suplem1pr  11093  wuncn  11211
  Copyright terms: Public domain W3C validator