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Theorem prnmadd 9857
Description: A positive real has no largest member. Addition version. (Contributed by NM, 7-Apr-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prnmadd ((𝐴P𝐵𝐴) → ∃𝑥(𝐵 +Q 𝑥) ∈ 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem prnmadd
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 prnmax 9855 . 2 ((𝐴P𝐵𝐴) → ∃𝑦𝐴 𝐵 <Q 𝑦)
2 ltrelnq 9786 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 5202 . . . . . 6 (𝐵 <Q 𝑦 → (𝐵Q𝑦Q))
43simprd 478 . . . . 5 (𝐵 <Q 𝑦𝑦Q)
5 ltexnq 9835 . . . . . 6 (𝑦Q → (𝐵 <Q 𝑦 ↔ ∃𝑥(𝐵 +Q 𝑥) = 𝑦))
65biimpcd 239 . . . . 5 (𝐵 <Q 𝑦 → (𝑦Q → ∃𝑥(𝐵 +Q 𝑥) = 𝑦))
74, 6mpd 15 . . . 4 (𝐵 <Q 𝑦 → ∃𝑥(𝐵 +Q 𝑥) = 𝑦)
8 eleq1a 2725 . . . . 5 (𝑦𝐴 → ((𝐵 +Q 𝑥) = 𝑦 → (𝐵 +Q 𝑥) ∈ 𝐴))
98eximdv 1886 . . . 4 (𝑦𝐴 → (∃𝑥(𝐵 +Q 𝑥) = 𝑦 → ∃𝑥(𝐵 +Q 𝑥) ∈ 𝐴))
107, 9syl5 34 . . 3 (𝑦𝐴 → (𝐵 <Q 𝑦 → ∃𝑥(𝐵 +Q 𝑥) ∈ 𝐴))
1110rexlimiv 3056 . 2 (∃𝑦𝐴 𝐵 <Q 𝑦 → ∃𝑥(𝐵 +Q 𝑥) ∈ 𝐴)
121, 11syl 17 1 ((𝐴P𝐵𝐴) → ∃𝑥(𝐵 +Q 𝑥) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942   class class class wbr 4685  (class class class)co 6690  Qcnq 9712   +Q cplq 9715   <Q cltq 9718  Pcnp 9719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-ltnq 9778  df-np 9841
This theorem is referenced by:  ltexprlem1  9896  ltexprlem7  9902
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