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Mirrors > Home > ILE Home > Th. List > recexprlemopu | GIF version |
Description: The upper cut of 𝐵 is open. Lemma for recexpr 7414. (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 |
---|---|
recexprlemopu | ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recexpr.1 | . . . 4 ⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 | |
2 | 1 | recexprlemelu 7399 | . . 3 ⊢ (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) |
3 | ltbtwnnqq 7191 | . . . . . 6 ⊢ (𝑦 <Q 𝑟 ↔ ∃𝑞 ∈ Q (𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) | |
4 | 3 | biimpi 119 | . . . . 5 ⊢ (𝑦 <Q 𝑟 → ∃𝑞 ∈ Q (𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) |
5 | simplr 504 | . . . . . . . 8 ⊢ (((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑞 <Q 𝑟) | |
6 | 19.8a 1554 | . . . . . . . . . 10 ⊢ ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) | |
7 | 1 | recexprlemelu 7399 | . . . . . . . . . 10 ⊢ (𝑞 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) |
8 | 6, 7 | sylibr 133 | . . . . . . . . 9 ⊢ ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑞 ∈ (2nd ‘𝐵)) |
9 | 8 | adantlr 468 | . . . . . . . 8 ⊢ (((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑞 ∈ (2nd ‘𝐵)) |
10 | 5, 9 | jca 304 | . . . . . . 7 ⊢ (((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
11 | 10 | expcom 115 | . . . . . 6 ⊢ ((*Q‘𝑦) ∈ (1st ‘𝐴) → ((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))) |
12 | 11 | reximdv 2510 | . . . . 5 ⊢ ((*Q‘𝑦) ∈ (1st ‘𝐴) → (∃𝑞 ∈ Q (𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))) |
13 | 4, 12 | mpan9 279 | . . . 4 ⊢ ((𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
14 | 13 | exlimiv 1562 | . . 3 ⊢ (∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
15 | 2, 14 | sylbi 120 | . 2 ⊢ (𝑟 ∈ (2nd ‘𝐵) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
16 | 15 | 3ad2ant3 989 | 1 ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 947 = wceq 1316 ∃wex 1453 ∈ wcel 1465 {cab 2103 ∃wrex 2394 〈cop 3500 class class class wbr 3899 ‘cfv 5093 1st c1st 6004 2nd c2nd 6005 Qcnq 7056 *Qcrq 7060 <Q cltq 7061 Pcnp 7067 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 |
This theorem is referenced by: recexprlemrnd 7405 |
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