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Theorem prnmax 9764
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 2686 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
21anbi2d 739 . . . 4 (𝑦 = 𝐵 → ((𝐴P𝑦𝐴) ↔ (𝐴P𝐵𝐴)))
3 breq1 4618 . . . . 5 (𝑦 = 𝐵 → (𝑦 <Q 𝑥𝐵 <Q 𝑥))
43rexbidv 3045 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 <Q 𝑥 ↔ ∃𝑥𝐴 𝐵 <Q 𝑥))
52, 4imbi12d 334 . . 3 (𝑦 = 𝐵 → (((𝐴P𝑦𝐴) → ∃𝑥𝐴 𝑦 <Q 𝑥) ↔ ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)))
6 elnpi 9757 . . . . . 6 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑦𝐴 (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥)))
76simprbi 480 . . . . 5 (𝐴P → ∀𝑦𝐴 (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥))
87r19.21bi 2927 . . . 4 ((𝐴P𝑦𝐴) → (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥))
98simprd 479 . . 3 ((𝐴P𝑦𝐴) → ∃𝑥𝐴 𝑦 <Q 𝑥)
105, 9vtoclg 3252 . 2 (𝐵𝐴 → ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥))
1110anabsi7 859 1 ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  wpss 3557  c0 3893   class class class wbr 4615  Qcnq 9621   <Q cltq 9627  Pcnp 9628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-np 9750
This theorem is referenced by:  npomex  9765  prnmadd  9766  genpnmax  9776  1idpr  9798  ltexprlem4  9808  reclem3pr  9818  suplem1pr  9821
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