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Mirrors > Home > ILE Home > Th. List > recexprlemopl | GIF version |
Description: The lower cut of 𝐵 is open. Lemma for recexpr 7639. (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 7623 | . . 3 ⊢ (𝑞 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))) |
3 | ltbtwnnqq 7416 | . . . . . 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 7623 | . . . . . . . . . 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 3597 class class class wbr 4005 ‘cfv 5218 1st c1st 6141 2nd c2nd 6142 Qcnq 7281 *Qcrq 7285 <Q cltq 7286 Pcnp 7292 |
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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 |
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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-eprel 4291 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-1o 6419 df-oadd 6423 df-omul 6424 df-er 6537 df-ec 6539 df-qs 6543 df-ni 7305 df-pli 7306 df-mi 7307 df-lti 7308 df-plpq 7345 df-mpq 7346 df-enq 7348 df-nqqs 7349 df-plqqs 7350 df-mqqs 7351 df-1nqqs 7352 df-rq 7353 df-ltnqqs 7354 |
This theorem is referenced by: recexprlemrnd 7630 |
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