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Theorem prn0 10400
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 10399 . . 3 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
2 simpl2 1189 . . 3 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴)
31, 2sylbi 220 . 2 (𝐴P → ∅ ⊊ 𝐴)
4 0pss 4368 . 2 (∅ ⊊ 𝐴𝐴 ≠ ∅)
53, 4sylib 221 1 (𝐴P𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wal 1536  wcel 2114  wne 3011  wral 3130  wrex 3131  Vcvv 3469  wpss 3909  c0 4265   class class class wbr 5042  Qcnq 10263   <Q cltq 10269  Pcnp 10270
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ne 3012  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-np 10392
This theorem is referenced by:  0npr  10403  npomex  10407  genpn0  10414  prlem934  10444  ltaddpr  10445  prlem936  10458  reclem2pr  10459  suplem1pr  10463
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