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Mirrors > Home > ILE Home > Th. List > prnmaddl | GIF version |
Description: A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.) |
Ref | Expression |
---|---|
prnmaddl | ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐵 ∈ 𝐿) → ∃𝑥 ∈ Q (𝐵 +Q 𝑥) ∈ 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prnmaxl 7101 | . 2 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐵 ∈ 𝐿) → ∃𝑦 ∈ 𝐿 𝐵 <Q 𝑦) | |
2 | ltexnqi 7022 | . . . 4 ⊢ (𝐵 <Q 𝑦 → ∃𝑥 ∈ Q (𝐵 +Q 𝑥) = 𝑦) | |
3 | eleq1a 2160 | . . . . 5 ⊢ (𝑦 ∈ 𝐿 → ((𝐵 +Q 𝑥) = 𝑦 → (𝐵 +Q 𝑥) ∈ 𝐿)) | |
4 | 3 | reximdv 2475 | . . . 4 ⊢ (𝑦 ∈ 𝐿 → (∃𝑥 ∈ Q (𝐵 +Q 𝑥) = 𝑦 → ∃𝑥 ∈ Q (𝐵 +Q 𝑥) ∈ 𝐿)) |
5 | 2, 4 | syl5 32 | . . 3 ⊢ (𝑦 ∈ 𝐿 → (𝐵 <Q 𝑦 → ∃𝑥 ∈ Q (𝐵 +Q 𝑥) ∈ 𝐿)) |
6 | 5 | rexlimiv 2484 | . 2 ⊢ (∃𝑦 ∈ 𝐿 𝐵 <Q 𝑦 → ∃𝑥 ∈ Q (𝐵 +Q 𝑥) ∈ 𝐿) |
7 | 1, 6 | syl 14 | 1 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐵 ∈ 𝐿) → ∃𝑥 ∈ Q (𝐵 +Q 𝑥) ∈ 𝐿) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 ∃wrex 2361 〈cop 3453 class class class wbr 3851 (class class class)co 5666 Qcnq 6893 +Q cplq 6895 <Q cltq 6898 Pcnp 6904 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-eprel 4125 df-id 4129 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-1o 6195 df-oadd 6199 df-omul 6200 df-er 6306 df-ec 6308 df-qs 6312 df-ni 6917 df-pli 6918 df-mi 6919 df-lti 6920 df-plpq 6957 df-mpq 6958 df-enq 6960 df-nqqs 6961 df-plqqs 6962 df-mqqs 6963 df-1nqqs 6964 df-ltnqqs 6966 df-inp 7079 |
This theorem is referenced by: ltexprlemrl 7223 addcanprleml 7227 |
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