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Mirrors > Home > ILE Home > Th. List > recexprlemopu | GIF version |
Description: The upper cut of 𝐵 is open. Lemma for recexpr 7570. (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 7555 | . . 3 ⊢ (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) |
3 | ltbtwnnqq 7347 | . . . . . 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 520 | . . . . . . . 8 ⊢ (((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑞 <Q 𝑟) | |
6 | 19.8a 1577 | . . . . . . . . . 10 ⊢ ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) | |
7 | 1 | recexprlemelu 7555 | . . . . . . . . . 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 469 | . . . . . . . 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 2565 | . . . . 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 1585 | . . 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 1009 | 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 967 = wceq 1342 ∃wex 1479 ∈ wcel 2135 {cab 2150 ∃wrex 2443 〈cop 3573 class class class wbr 3976 ‘cfv 5182 1st c1st 6098 2nd c2nd 6099 Qcnq 7212 *Qcrq 7216 <Q cltq 7217 Pcnp 7223 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 |
This theorem is referenced by: recexprlemrnd 7561 |
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