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Theorem prn0 10012
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 10011 . . 3 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
2 simpl2 1228 . . 3 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴)
31, 2sylbi 207 . 2 (𝐴P → ∅ ⊊ 𝐴)
4 0pss 4155 . 2 (∅ ⊊ 𝐴𝐴 ≠ ∅)
53, 4sylib 208 1 (𝐴P𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070  wal 1628  wcel 2144  wne 2942  wral 3060  wrex 3061  Vcvv 3349  wpss 3722  c0 4061   class class class wbr 4784  Qcnq 9875   <Q cltq 9881  Pcnp 9882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-v 3351  df-dif 3724  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-np 10004
This theorem is referenced by:  0npr  10015  npomex  10019  genpn0  10026  prlem934  10056  ltaddpr  10057  prlem936  10070  reclem2pr  10071  suplem1pr  10075
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