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Mirrors > Home > ILE Home > Th. List > recexprlemopu | GIF version |
Description: The upper cut of 𝐵 is open. Lemma for recexpr 7625. (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 7610 | . . 3 ⊢ (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) |
3 | ltbtwnnqq 7402 | . . . . . 6 ⊢ (𝑦 <Q 𝑟 ↔ ∃𝑞 ∈ Q (𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) | |
4 | 3 | biimpi 120 | . . . . 5 ⊢ (𝑦 <Q 𝑟 → ∃𝑞 ∈ Q (𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) |
5 | simplr 528 | . . . . . . . 8 ⊢ (((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑞 <Q 𝑟) | |
6 | 19.8a 1590 | . . . . . . . . . 10 ⊢ ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) | |
7 | 1 | recexprlemelu 7610 | . . . . . . . . . 10 ⊢ (𝑞 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) |
8 | 6, 7 | sylibr 134 | . . . . . . . . 9 ⊢ ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑞 ∈ (2nd ‘𝐵)) |
9 | 8 | adantlr 477 | . . . . . . . 8 ⊢ (((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑞 ∈ (2nd ‘𝐵)) |
10 | 5, 9 | jca 306 | . . . . . . 7 ⊢ (((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
11 | 10 | expcom 116 | . . . . . 6 ⊢ ((*Q‘𝑦) ∈ (1st ‘𝐴) → ((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))) |
12 | 11 | reximdv 2578 | . . . . 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 281 | . . . 4 ⊢ ((𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
14 | 13 | exlimiv 1598 | . . 3 ⊢ (∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
15 | 2, 14 | sylbi 121 | . 2 ⊢ (𝑟 ∈ (2nd ‘𝐵) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
16 | 15 | 3ad2ant3 1020 | 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 104 ∧ w3a 978 = wceq 1353 ∃wex 1492 ∈ wcel 2148 {cab 2163 ∃wrex 2456 〈cop 3594 class class class wbr 4000 ‘cfv 5212 1st c1st 6133 2nd c2nd 6134 Qcnq 7267 *Qcrq 7271 <Q cltq 7272 Pcnp 7278 |
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 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4286 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-1o 6411 df-oadd 6415 df-omul 6416 df-er 6529 df-ec 6531 df-qs 6535 df-ni 7291 df-pli 7292 df-mi 7293 df-lti 7294 df-plpq 7331 df-mpq 7332 df-enq 7334 df-nqqs 7335 df-plqqs 7336 df-mqqs 7337 df-1nqqs 7338 df-rq 7339 df-ltnqqs 7340 |
This theorem is referenced by: recexprlemrnd 7616 |
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