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Mirrors > Home > ILE Home > Th. List > recexprlemupu | GIF version |
Description: The upper cut of 𝐵 is upper. Lemma for recexpr 7470. (Contributed by Jim Kingdon, 28-Dec-2019.) |
Ref | Expression |
---|---|
recexpr.1 | ⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 |
Ref | Expression |
---|---|
recexprlemupu | ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltsonq 7230 | . . . . . . . . 9 ⊢ <Q Or Q | |
2 | ltrelnq 7197 | . . . . . . . . 9 ⊢ <Q ⊆ (Q × Q) | |
3 | 1, 2 | sotri 4942 | . . . . . . . 8 ⊢ ((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → 𝑦 <Q 𝑟) |
4 | 3 | expcom 115 | . . . . . . 7 ⊢ (𝑞 <Q 𝑟 → (𝑦 <Q 𝑞 → 𝑦 <Q 𝑟)) |
5 | 4 | anim1d 334 | . . . . . 6 ⊢ (𝑞 <Q 𝑟 → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → (𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))) |
6 | 5 | eximdv 1853 | . . . . 5 ⊢ (𝑞 <Q 𝑟 → (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))) |
7 | recexpr.1 | . . . . . 6 ⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 | |
8 | 7 | recexprlemelu 7455 | . . . . 5 ⊢ (𝑞 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) |
9 | 7 | recexprlemelu 7455 | . . . . 5 ⊢ (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) |
10 | 6, 8, 9 | 3imtr4g 204 | . . . 4 ⊢ (𝑞 <Q 𝑟 → (𝑞 ∈ (2nd ‘𝐵) → 𝑟 ∈ (2nd ‘𝐵))) |
11 | 10 | imp 123 | . . 3 ⊢ ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐵)) |
12 | 11 | rexlimivw 2548 | . 2 ⊢ (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐵)) |
13 | 12 | a1i 9 | 1 ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∃wex 1469 ∈ wcel 1481 {cab 2126 ∃wrex 2418 〈cop 3535 class class class wbr 3937 ‘cfv 5131 1st c1st 6044 2nd c2nd 6045 Qcnq 7112 *Qcrq 7116 <Q cltq 7117 Pcnp 7123 |
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-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-mi 7138 df-lti 7139 df-enq 7179 df-nqqs 7180 df-ltnqqs 7185 |
This theorem is referenced by: recexprlemrnd 7461 |
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