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Theorem prnmax 10029
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 2827 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
21anbi2d 742 . . . 4 (𝑦 = 𝐵 → ((𝐴P𝑦𝐴) ↔ (𝐴P𝐵𝐴)))
3 breq1 4807 . . . . 5 (𝑦 = 𝐵 → (𝑦 <Q 𝑥𝐵 <Q 𝑥))
43rexbidv 3190 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 <Q 𝑥 ↔ ∃𝑥𝐴 𝐵 <Q 𝑥))
52, 4imbi12d 333 . . 3 (𝑦 = 𝐵 → (((𝐴P𝑦𝐴) → ∃𝑥𝐴 𝑦 <Q 𝑥) ↔ ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)))
6 elnpi 10022 . . . . . 6 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑦𝐴 (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥)))
76simprbi 483 . . . . 5 (𝐴P → ∀𝑦𝐴 (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥))
87r19.21bi 3070 . . . 4 ((𝐴P𝑦𝐴) → (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥))
98simprd 482 . . 3 ((𝐴P𝑦𝐴) → ∃𝑥𝐴 𝑦 <Q 𝑥)
105, 9vtoclg 3406 . 2 (𝐵𝐴 → ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥))
1110anabsi7 895 1 ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wal 1630   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  wpss 3716  c0 4058   class class class wbr 4804  Qcnq 9886   <Q cltq 9892  Pcnp 9893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-np 10015
This theorem is referenced by:  npomex  10030  prnmadd  10031  genpnmax  10041  1idpr  10063  ltexprlem4  10073  reclem3pr  10083  suplem1pr  10086
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