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Mirrors > Home > ILE Home > Th. List > recexprlemopl | GIF version |
Description: The lower cut of 𝐵 is open. Lemma for recexpr 7600. (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 |
---|---|
recexprlemopl | ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1st ‘𝐵)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘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 | recexprlemell 7584 | . . 3 ⊢ (𝑞 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))) |
3 | ltbtwnnqq 7377 | . . . . . 6 ⊢ (𝑞 <Q 𝑦 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦)) | |
4 | 3 | biimpi 119 | . . . . 5 ⊢ (𝑞 <Q 𝑦 → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦)) |
5 | simpll 524 | . . . . . . . 8 ⊢ (((𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦) ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑞 <Q 𝑟) | |
6 | 19.8a 1583 | . . . . . . . . . 10 ⊢ ((𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → ∃𝑦(𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))) | |
7 | 1 | recexprlemell 7584 | . . . . . . . . . 10 ⊢ (𝑟 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))) |
8 | 6, 7 | sylibr 133 | . . . . . . . . 9 ⊢ ((𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑟 ∈ (1st ‘𝐵)) |
9 | 8 | adantll 473 | . . . . . . . 8 ⊢ (((𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦) ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑟 ∈ (1st ‘𝐵)) |
10 | 5, 9 | jca 304 | . . . . . . 7 ⊢ (((𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦) ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) |
11 | 10 | expcom 115 | . . . . . 6 ⊢ ((*Q‘𝑦) ∈ (2nd ‘𝐴) → ((𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦) → (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)))) |
12 | 11 | reximdv 2571 | . . . . 5 ⊢ ((*Q‘𝑦) ∈ (2nd ‘𝐴) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)))) |
13 | 4, 12 | mpan9 279 | . . . 4 ⊢ ((𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) |
14 | 13 | exlimiv 1591 | . . 3 ⊢ (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) |
15 | 2, 14 | sylbi 120 | . 2 ⊢ (𝑞 ∈ (1st ‘𝐵) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) |
16 | 15 | 3ad2ant3 1015 | 1 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1st ‘𝐵)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 = wceq 1348 ∃wex 1485 ∈ wcel 2141 {cab 2156 ∃wrex 2449 〈cop 3586 class class class wbr 3989 ‘cfv 5198 1st c1st 6117 2nd c2nd 6118 Qcnq 7242 *Qcrq 7246 <Q cltq 7247 Pcnp 7253 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 |
This theorem is referenced by: recexprlemrnd 7591 |
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