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Theorem recexprlemopl 7381

Description: The lower cut of 𝐵 is open. Lemma for recexpr 7394. (Contributed by Jim Kingdon, 28-Dec-2019.)

**Hypothesis**
Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩

**Assertion**
Ref	Expression
recexprlemopl	⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐵)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))

Distinct variable groups: 𝑟,𝑞,𝑥,𝑦,𝐴 𝐵,𝑞,𝑟,𝑥,𝑦

**Proof of Theorem recexprlemopl**
Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
2	1	recexprlemell 7378	. . 3 ⊢ (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
3		ltbtwnnqq 7171	. . . . . 6 ⊢ (𝑞 <_Q 𝑦 ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦))
4	3	biimpi 119	. . . . 5 ⊢ (𝑞 <_Q 𝑦 → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦))
5		simpll 501	. . . . . . . 8 ⊢ (((𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑞 <_Q 𝑟)
6		19.8a 1552	. . . . . . . . . 10 ⊢ ((𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) → ∃𝑦(𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)))
7	1	recexprlemell 7378	. . . . . . . . . 10 ⊢ (𝑟 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑟 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
8	6, 7	sylibr 133	. . . . . . . . 9 ⊢ ((𝑟 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑟 ∈ (1^st ‘𝐵))
9	8	adantll 465	. . . . . . . 8 ⊢ (((𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑟 ∈ (1^st ‘𝐵))
10	5, 9	jca 302	. . . . . . 7 ⊢ (((𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))
11	10	expcom 115	. . . . . 6 ⊢ ((*_Q‘𝑦) ∈ (2^nd ‘𝐴) → ((𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) → (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))))
12	11	reximdv 2507	. . . . 5 ⊢ ((*_Q‘𝑦) ∈ (2^nd ‘𝐴) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))))
13	4, 12	mpan9 277	. . . 4 ⊢ ((𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))
14	13	exlimiv 1560	. . 3 ⊢ (∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))
15	2, 14	sylbi 120	. 2 ⊢ (𝑞 ∈ (1^st ‘𝐵) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))
16	15	3ad2ant3 987	1 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐵)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 945 = wceq 1314 ∃wex 1451 ∈ wcel 1463 {cab 2101 ∃wrex 2391 ⟨cop 3496 class class class wbr 3895 ‘cfv 5081 1^st c1st 5990 2^nd c2nd 5991 Qcnq 7036 *_Qcrq 7040 <_Q cltq 7041 Pcnp 7047

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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-iinf 4462

This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-tr 3987 df-eprel 4171 df-id 4175 df-po 4178 df-iso 4179 df-iord 4248 df-on 4250 df-suc 4253 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-recs 6156 df-irdg 6221 df-1o 6267 df-oadd 6271 df-omul 6272 df-er 6383 df-ec 6385 df-qs 6389 df-ni 7060 df-pli 7061 df-mi 7062 df-lti 7063 df-plpq 7100 df-mpq 7101 df-enq 7103 df-nqqs 7104 df-plqqs 7105 df-mqqs 7106 df-1nqqs 7107 df-rq 7108 df-ltnqqs 7109

This theorem is referenced by: recexprlemrnd 7385

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