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Theorem prnmax 10989
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 2815 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
21anbi2d 628 . . . 4 (𝑦 = 𝐵 → ((𝐴P𝑦𝐴) ↔ (𝐴P𝐵𝐴)))
3 breq1 5144 . . . . 5 (𝑦 = 𝐵 → (𝑦 <Q 𝑥𝐵 <Q 𝑥))
43rexbidv 3172 . . . 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 10982 . . . . . 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 3242 . . . 4 ((𝐴P𝑦𝐴) → (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥))
98simprd 495 . . 3 ((𝐴P𝑦𝐴) → ∃𝑥𝐴 𝑦 <Q 𝑥)
105, 9vtoclg 3537 . 2 (𝐵𝐴 → ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥))
1110anabsi7 668 1 ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wal 1531   = wceq 1533  wcel 2098  wral 3055  wrex 3064  Vcvv 3468  wpss 3944  c0 4317   class class class wbr 5141  Qcnq 10846   <Q cltq 10852  Pcnp 10853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-np 10975
This theorem is referenced by:  npomex  10990  prnmadd  10991  genpnmax  11001  1idpr  11023  ltexprlem4  11033  reclem3pr  11043  suplem1pr  11046
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