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Theorem recexprlemlol 7993

Description: The lower cut of 𝐵 is lower. Lemma for recexpr 8005. (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
recexprlemlol	⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)) → 𝑞 ∈ (1^st ‘𝐵)))

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

**Proof of Theorem recexprlemlol**
Step	Hyp	Ref	Expression
1		ltsonq 7765	. . . . . . . . 9 ⊢ <_Q Or Q
2		ltrelnq 7732	. . . . . . . . 9 ⊢ <_Q ⊆ (Q × Q)
3	1, 2	sotri 5183	. . . . . . . 8 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) → 𝑞 <_Q 𝑦)
4	3	ex 115	. . . . . . 7 ⊢ (𝑞 <_Q 𝑟 → (𝑟 <_Q 𝑦 → 𝑞 <_Q 𝑦))
5	4	anim1d 336	. . . . . 6 ⊢ (𝑞 <_Q 𝑟 → ((𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) → (𝑞 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))))
6	5	eximdv 1933	. . . . 5 ⊢ (𝑞 <_Q 𝑟 → (∃𝑦(𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) → ∃𝑦(𝑞 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))))
7		recexpr.1	. . . . . 6 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
8	7	recexprlemell 7989	. . . . 5 ⊢ (𝑟 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑟 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
9	7	recexprlemell 7989	. . . . 5 ⊢ (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
10	6, 8, 9	3imtr4g 205	. . . 4 ⊢ (𝑞 <_Q 𝑟 → (𝑟 ∈ (1^st ‘𝐵) → 𝑞 ∈ (1^st ‘𝐵)))
11	10	imp 124	. . 3 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)) → 𝑞 ∈ (1^st ‘𝐵))
12	11	rexlimivw 2664	. 2 ⊢ (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)) → 𝑞 ∈ (1^st ‘𝐵))
13	12	a1i 9	1 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)) → 𝑞 ∈ (1^st ‘𝐵)))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∃wex 1545 ∈ wcel 2209 {cab 2224 ∃wrex 2529 ⟨cop 3712 class class class wbr 4130 ‘cfv 5377 1^st c1st 6372 2^nd c2nd 6373 Qcnq 7647 *_Qcrq 7651 <_Q cltq 7652 Pcnp 7658

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-eprel 4434 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-oadd 6691 df-omul 6692 df-er 6807 df-ec 6809 df-qs 6813 df-ni 7671 df-mi 7673 df-lti 7674 df-enq 7714 df-nqqs 7715 df-ltnqqs 7720

This theorem is used by: recexprlemrnd 7996

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