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Theorem prnmax 11019
Description: A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prnmax ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem prnmax
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2817 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
21anbi2d 629 . . . 4 (𝑦 = 𝐵 → ((𝐴P𝑦𝐴) ↔ (𝐴P𝐵𝐴)))
3 breq1 5151 . . . . 5 (𝑦 = 𝐵 → (𝑦 <Q 𝑥𝐵 <Q 𝑥))
43rexbidv 3175 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 <Q 𝑥 ↔ ∃𝑥𝐴 𝐵 <Q 𝑥))
52, 4imbi12d 344 . . 3 (𝑦 = 𝐵 → (((𝐴P𝑦𝐴) → ∃𝑥𝐴 𝑦 <Q 𝑥) ↔ ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)))
6 elnpi 11012 . . . . . 6 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑦𝐴 (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥)))
76simprbi 496 . . . . 5 (𝐴P → ∀𝑦𝐴 (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥))
87r19.21bi 3245 . . . 4 ((𝐴P𝑦𝐴) → (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥))
98simprd 495 . . 3 ((𝐴P𝑦𝐴) → ∃𝑥𝐴 𝑦 <Q 𝑥)
105, 9vtoclg 3540 . 2 (𝐵𝐴 → ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥))
1110anabsi7 670 1 ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085  wal 1532   = wceq 1534  wcel 2099  wral 3058  wrex 3067  Vcvv 3471  wpss 3948  c0 4323   class class class wbr 5148  Qcnq 10876   <Q cltq 10882  Pcnp 10883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-np 11005
This theorem is referenced by:  npomex  11020  prnmadd  11021  genpnmax  11031  1idpr  11053  ltexprlem4  11063  reclem3pr  11073  suplem1pr  11076
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