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Theorem prnmadd 9763
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 9761 . 2 ((𝐴P𝐵𝐴) → ∃𝑦𝐴 𝐵 <Q 𝑦)
2 ltrelnq 9692 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 5128 . . . . . 6 (𝐵 <Q 𝑦 → (𝐵Q𝑦Q))
43simprd 479 . . . . 5 (𝐵 <Q 𝑦𝑦Q)
5 ltexnq 9741 . . . . . 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 2693 . . . . 5 (𝑦𝐴 → ((𝐵 +Q 𝑥) = 𝑦 → (𝐵 +Q 𝑥) ∈ 𝐴))
98eximdv 1843 . . . 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 3020 . 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 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908   class class class wbr 4613  (class class class)co 6604  Qcnq 9618   +Q cplq 9621   <Q cltq 9624  Pcnp 9625
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-omul 7510  df-er 7687  df-ni 9638  df-pli 9639  df-mi 9640  df-lti 9641  df-plpq 9674  df-mpq 9675  df-ltpq 9676  df-enq 9677  df-nq 9678  df-erq 9679  df-plq 9680  df-mq 9681  df-1nq 9682  df-ltnq 9684  df-np 9747
This theorem is referenced by:  ltexprlem1  9802  ltexprlem7  9808
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