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Theorem prn0 9796
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 9795 . . 3 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
2 simpl2 1063 . . 3 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴)
31, 2sylbi 207 . 2 (𝐴P → ∅ ⊊ 𝐴)
4 0pss 4004 . 2 (∅ ⊊ 𝐴𝐴 ≠ ∅)
53, 4sylib 208 1 (𝐴P𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wal 1479  wcel 1988  wne 2791  wral 2909  wrex 2910  Vcvv 3195  wpss 3568  c0 3907   class class class wbr 4644  Qcnq 9659   <Q cltq 9665  Pcnp 9666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-np 9788
This theorem is referenced by:  0npr  9799  npomex  9803  genpn0  9810  prlem934  9840  ltaddpr  9841  prlem936  9854  reclem2pr  9855  suplem1pr  9859
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