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Mirrors > Home > ILE Home > Th. List > recexprlemlol | GIF version |
Description: The lower cut of 𝐵 is lower. Lemma for recexpr 7553. (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 |
---|---|
recexprlemlol | ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)) → 𝑞 ∈ (1st ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltsonq 7313 | . . . . . . . . 9 ⊢ <Q Or Q | |
2 | ltrelnq 7280 | . . . . . . . . 9 ⊢ <Q ⊆ (Q × Q) | |
3 | 1, 2 | sotri 4980 | . . . . . . . 8 ⊢ ((𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝑦) → 𝑞 <Q 𝑦) |
4 | 3 | ex 114 | . . . . . . 7 ⊢ (𝑞 <Q 𝑟 → (𝑟 <Q 𝑦 → 𝑞 <Q 𝑦)) |
5 | 4 | anim1d 334 | . . . . . 6 ⊢ (𝑞 <Q 𝑟 → ((𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → (𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))) |
6 | 5 | eximdv 1860 | . . . . 5 ⊢ (𝑞 <Q 𝑟 → (∃𝑦(𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))) |
7 | recexpr.1 | . . . . . 6 ⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 | |
8 | 7 | recexprlemell 7537 | . . . . 5 ⊢ (𝑟 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑟 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))) |
9 | 7 | recexprlemell 7537 | . . . . 5 ⊢ (𝑞 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))) |
10 | 6, 8, 9 | 3imtr4g 204 | . . . 4 ⊢ (𝑞 <Q 𝑟 → (𝑟 ∈ (1st ‘𝐵) → 𝑞 ∈ (1st ‘𝐵))) |
11 | 10 | imp 123 | . . 3 ⊢ ((𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)) → 𝑞 ∈ (1st ‘𝐵)) |
12 | 11 | rexlimivw 2570 | . 2 ⊢ (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)) → 𝑞 ∈ (1st ‘𝐵)) |
13 | 12 | a1i 9 | 1 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)) → 𝑞 ∈ (1st ‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∃wex 1472 ∈ wcel 2128 {cab 2143 ∃wrex 2436 〈cop 3563 class class class wbr 3965 ‘cfv 5169 1st c1st 6083 2nd c2nd 6084 Qcnq 7195 *Qcrq 7199 <Q cltq 7200 Pcnp 7206 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-oadd 6364 df-omul 6365 df-er 6477 df-ec 6479 df-qs 6483 df-ni 7219 df-mi 7221 df-lti 7222 df-enq 7262 df-nqqs 7263 df-ltnqqs 7268 |
This theorem is referenced by: recexprlemrnd 7544 |
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