![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > suplocsr | GIF version |
Description: An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.) |
Ref | Expression |
---|---|
suplocsr.m | ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
suplocsr.ub | ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) |
suplocsr.loc | ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) |
Ref | Expression |
---|---|
suplocsr | ⊢ (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suplocsr.m | . . 3 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) | |
2 | eleq1w 2201 | . . . 4 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴)) | |
3 | 2 | cbvexv 1891 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴) |
4 | 1, 3 | sylib 121 | . 2 ⊢ (𝜑 → ∃𝑎 𝑎 ∈ 𝐴) |
5 | opeq1 3713 | . . . . . . 7 ⊢ (𝑏 = 𝑐 → 〈𝑏, 1P〉 = 〈𝑐, 1P〉) | |
6 | 5 | eceq1d 6473 | . . . . . 6 ⊢ (𝑏 = 𝑐 → [〈𝑏, 1P〉] ~R = [〈𝑐, 1P〉] ~R ) |
7 | 6 | oveq2d 5798 | . . . . 5 ⊢ (𝑏 = 𝑐 → (𝑎 +R [〈𝑏, 1P〉] ~R ) = (𝑎 +R [〈𝑐, 1P〉] ~R )) |
8 | 7 | eleq1d 2209 | . . . 4 ⊢ (𝑏 = 𝑐 → ((𝑎 +R [〈𝑏, 1P〉] ~R ) ∈ 𝐴 ↔ (𝑎 +R [〈𝑐, 1P〉] ~R ) ∈ 𝐴)) |
9 | 8 | cbvrabv 2688 | . . 3 ⊢ {𝑏 ∈ P ∣ (𝑎 +R [〈𝑏, 1P〉] ~R ) ∈ 𝐴} = {𝑐 ∈ P ∣ (𝑎 +R [〈𝑐, 1P〉] ~R ) ∈ 𝐴} |
10 | suplocsr.ub | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) | |
11 | ltrelsr 7570 | . . . . . . . . . 10 ⊢ <R ⊆ (R × R) | |
12 | 11 | brel 4599 | . . . . . . . . 9 ⊢ (𝑦 <R 𝑥 → (𝑦 ∈ R ∧ 𝑥 ∈ R)) |
13 | 12 | simpld 111 | . . . . . . . 8 ⊢ (𝑦 <R 𝑥 → 𝑦 ∈ R) |
14 | 13 | ralimi 2498 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ R) |
15 | dfss3 3092 | . . . . . . 7 ⊢ (𝐴 ⊆ R ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ R) | |
16 | 14, 15 | sylibr 133 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → 𝐴 ⊆ R) |
17 | 16 | rexlimivw 2548 | . . . . 5 ⊢ (∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → 𝐴 ⊆ R) |
18 | 10, 17 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ R) |
19 | 18 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐴 ⊆ R) |
20 | simpr 109 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴) | |
21 | 10 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) |
22 | suplocsr.loc | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) | |
23 | 22 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) |
24 | 9, 19, 20, 21, 23 | suplocsrlem 7640 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) |
25 | 4, 24 | exlimddv 1871 | 1 ⊢ (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 ∃wex 1469 ∈ wcel 1481 ∀wral 2417 ∃wrex 2418 {crab 2421 ⊆ wss 3076 〈cop 3535 class class class wbr 3937 (class class class)co 5782 [cec 6435 Pcnp 7123 1Pc1p 7124 ~R cer 7128 Rcnr 7129 +R cplr 7133 <R cltr 7135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-i1p 7299 df-iplp 7300 df-imp 7301 df-iltp 7302 df-enr 7558 df-nr 7559 df-plr 7560 df-mr 7561 df-ltr 7562 df-0r 7563 df-1r 7564 df-m1r 7565 |
This theorem is referenced by: axpre-suploclemres 7733 |
Copyright terms: Public domain | W3C validator |