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Mirrors > Home > ILE Home > Th. List > recexprlemopu | GIF version |
Description: The upper cut of 𝐵 is open. Lemma for recexpr 7294. (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 7279 | . . 3 ⊢ (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) |
3 | ltbtwnnqq 7071 | . . . . . 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 498 | . . . . . . . 8 ⊢ (((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑞 <Q 𝑟) | |
6 | 19.8a 1534 | . . . . . . . . . 10 ⊢ ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) | |
7 | 1 | recexprlemelu 7279 | . . . . . . . . . 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 462 | . . . . . . . 8 ⊢ (((𝑦 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑞 ∈ (2nd ‘𝐵)) |
10 | 5, 9 | jca 301 | . . . . . . 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 2486 | . . . . 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 276 | . . . 4 ⊢ ((𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) |
14 | 13 | exlimiv 1541 | . . 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 969 | 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 927 = wceq 1296 ∃wex 1433 ∈ wcel 1445 {cab 2081 ∃wrex 2371 〈cop 3469 class class class wbr 3867 ‘cfv 5049 1st c1st 5947 2nd c2nd 5948 Qcnq 6936 *Qcrq 6940 <Q cltq 6941 Pcnp 6947 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-eprel 4140 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-irdg 6173 df-1o 6219 df-oadd 6223 df-omul 6224 df-er 6332 df-ec 6334 df-qs 6338 df-ni 6960 df-pli 6961 df-mi 6962 df-lti 6963 df-plpq 7000 df-mpq 7001 df-enq 7003 df-nqqs 7004 df-plqqs 7005 df-mqqs 7006 df-1nqqs 7007 df-rq 7008 df-ltnqqs 7009 |
This theorem is referenced by: recexprlemrnd 7285 |
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