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Theorem prn0 11029
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 11028 . . 3 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
2 simpl2 1193 . . 3 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴)
31, 2sylbi 217 . 2 (𝐴P → ∅ ⊊ 𝐴)
4 0pss 4447 . 2 (∅ ⊊ 𝐴𝐴 ≠ ∅)
53, 4sylib 218 1 (𝐴P𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wal 1538  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  wpss 3952  c0 4333   class class class wbr 5143  Qcnq 10892   <Q cltq 10898  Pcnp 10899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-ss 3968  df-pss 3971  df-nul 4334  df-np 11021
This theorem is referenced by:  0npr  11032  npomex  11036  genpn0  11043  prlem934  11073  ltaddpr  11074  prlem936  11087  reclem2pr  11088  suplem1pr  11092
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