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Theorem prn0 10099
Description: A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prn0 (𝐴P𝐴 ≠ ∅)

Proof of Theorem prn0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnpi 10098 . . 3 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
2 simpl2 1245 . . 3 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴)
31, 2sylbi 209 . 2 (𝐴P → ∅ ⊊ 𝐴)
4 0pss 4209 . 2 (∅ ⊊ 𝐴𝐴 ≠ ∅)
53, 4sylib 210 1 (𝐴P𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108  wal 1651  wcel 2157  wne 2971  wral 3089  wrex 3090  Vcvv 3385  wpss 3770  c0 4115   class class class wbr 4843  Qcnq 9962   <Q cltq 9968  Pcnp 9969
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 2777
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 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-np 10091
This theorem is referenced by:  0npr  10102  npomex  10106  genpn0  10113  prlem934  10143  ltaddpr  10144  prlem936  10157  reclem2pr  10158  suplem1pr  10162
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