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Mirrors > Home > ILE Home > Th. List > recexprlemopl | GIF version |
Description: The lower cut of 𝐵 is open. Lemma for recexpr 7634. (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 7618 | . . 3 ⊢ (𝑞 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))) |
3 | ltbtwnnqq 7411 | . . . . . 6 ⊢ (𝑞 <Q 𝑦 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦)) | |
4 | 3 | biimpi 120 | . . . . 5 ⊢ (𝑞 <Q 𝑦 → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦)) |
5 | simpll 527 | . . . . . . . 8 ⊢ (((𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦) ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑞 <Q 𝑟) | |
6 | 19.8a 1590 | . . . . . . . . . 10 ⊢ ((𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → ∃𝑦(𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))) | |
7 | 1 | recexprlemell 7618 | . . . . . . . . . 10 ⊢ (𝑟 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))) |
8 | 6, 7 | sylibr 134 | . . . . . . . . 9 ⊢ ((𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑟 ∈ (1st ‘𝐵)) |
9 | 8 | adantll 476 | . . . . . . . 8 ⊢ (((𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦) ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑟 ∈ (1st ‘𝐵)) |
10 | 5, 9 | jca 306 | . . . . . . 7 ⊢ (((𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦) ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) |
11 | 10 | expcom 116 | . . . . . 6 ⊢ ((*Q‘𝑦) ∈ (2nd ‘𝐴) → ((𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦) → (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)))) |
12 | 11 | reximdv 2578 | . . . . 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 281 | . . . 4 ⊢ ((𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) |
14 | 13 | exlimiv 1598 | . . 3 ⊢ (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) |
15 | 2, 14 | sylbi 121 | . 2 ⊢ (𝑞 ∈ (1st ‘𝐵) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) |
16 | 15 | 3ad2ant3 1020 | 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 104 ∧ w3a 978 = wceq 1353 ∃wex 1492 ∈ wcel 2148 {cab 2163 ∃wrex 2456 〈cop 3595 class class class wbr 4002 ‘cfv 5215 1st c1st 6136 2nd c2nd 6137 Qcnq 7276 *Qcrq 7280 <Q cltq 7281 Pcnp 7287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-eprel 4288 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-irdg 6368 df-1o 6414 df-oadd 6418 df-omul 6419 df-er 6532 df-ec 6534 df-qs 6538 df-ni 7300 df-pli 7301 df-mi 7302 df-lti 7303 df-plpq 7340 df-mpq 7341 df-enq 7343 df-nqqs 7344 df-plqqs 7345 df-mqqs 7346 df-1nqqs 7347 df-rq 7348 df-ltnqqs 7349 |
This theorem is referenced by: recexprlemrnd 7625 |
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