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Theorem aptiprlemu 6795
 Description: Lemma for aptipr 6796. (Contributed by Jim Kingdon, 28-Jan-2020.)
Assertion
Ref Expression
aptiprlemu ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (2nd𝐵) ⊆ (2nd𝐴))

Proof of Theorem aptiprlemu
Dummy variables 𝑓 𝑔 𝑠 𝑡 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6630 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 6644 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
31, 2sylan 271 . . . . 5 ((𝐵P𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
433ad2antl2 1078 . . . 4 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
5 simprr 492 . . . . . 6 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 <Q 𝑥)
6 ltexnqi 6564 . . . . . 6 (𝑠 <Q 𝑥 → ∃𝑡Q (𝑠 +Q 𝑡) = 𝑥)
75, 6syl 14 . . . . 5 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → ∃𝑡Q (𝑠 +Q 𝑡) = 𝑥)
8 simpl1 918 . . . . . . . 8 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝐴P)
98ad2antrr 465 . . . . . . 7 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝐴P)
10 simprl 491 . . . . . . 7 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝑡Q)
11 prop 6630 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 prarloc2 6659 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1311, 12sylan 271 . . . . . . 7 ((𝐴P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
149, 10, 13syl2anc 397 . . . . . 6 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
15 simpl2 919 . . . . . . . . . 10 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝐵P)
1615ad3antrrr 469 . . . . . . . . 9 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝐵P)
17 simpr 107 . . . . . . . . . 10 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝑥 ∈ (2nd𝐵))
1817ad3antrrr 469 . . . . . . . . 9 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑥 ∈ (2nd𝐵))
19 elprnqu 6637 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
201, 19sylan 271 . . . . . . . . 9 ((𝐵P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
2116, 18, 20syl2anc 397 . . . . . . . 8 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑥Q)
228ad3antrrr 469 . . . . . . . . . 10 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝐴P)
23 simprl 491 . . . . . . . . . 10 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑢 ∈ (1st𝐴))
24 elprnql 6636 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
2511, 24sylan 271 . . . . . . . . . 10 ((𝐴P𝑢 ∈ (1st𝐴)) → 𝑢Q)
2622, 23, 25syl2anc 397 . . . . . . . . 9 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑢Q)
2710adantr 265 . . . . . . . . 9 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑡Q)
28 addclnq 6530 . . . . . . . . 9 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
2926, 27, 28syl2anc 397 . . . . . . . 8 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑢 +Q 𝑡) ∈ Q)
30 nqtri3or 6551 . . . . . . . 8 ((𝑥Q ∧ (𝑢 +Q 𝑡) ∈ Q) → (𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥))
3121, 29, 30syl2anc 397 . . . . . . 7 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥))
3215adantr 265 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝐵P)
33 simprl 491 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 ∈ (2nd𝐵))
34 elprnqu 6637 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
351, 34sylan 271 . . . . . . . . . . . . . 14 ((𝐵P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
3632, 33, 35syl2anc 397 . . . . . . . . . . . . 13 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠Q)
3736ad3antrrr 469 . . . . . . . . . . . 12 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠Q)
3833ad3antrrr 469 . . . . . . . . . . . 12 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠 ∈ (2nd𝐵))
39 simplrr 496 . . . . . . . . . . . . . . . 16 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑠 +Q 𝑡) = 𝑥)
40 breq1 3794 . . . . . . . . . . . . . . . . 17 ((𝑠 +Q 𝑡) = 𝑥 → ((𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡) ↔ 𝑥 <Q (𝑢 +Q 𝑡)))
4140biimprd 151 . . . . . . . . . . . . . . . 16 ((𝑠 +Q 𝑡) = 𝑥 → (𝑥 <Q (𝑢 +Q 𝑡) → (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡)))
4239, 41syl 14 . . . . . . . . . . . . . . 15 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑥 <Q (𝑢 +Q 𝑡) → (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡)))
4342imp 119 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡))
44 ltanqg 6555 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
4544adantl 266 . . . . . . . . . . . . . . 15 ((((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
4626adantr 265 . . . . . . . . . . . . . . 15 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑢Q)
4727adantr 265 . . . . . . . . . . . . . . 15 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑡Q)
48 addcomnqg 6536 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4948adantl 266 . . . . . . . . . . . . . . 15 ((((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
5045, 37, 46, 47, 49caovord2d 5697 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝑠 <Q 𝑢 ↔ (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡)))
5143, 50mpbird 160 . . . . . . . . . . . . 13 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠 <Q 𝑢)
5222adantr 265 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝐴P)
5323adantr 265 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑢 ∈ (1st𝐴))
54 prcdnql 6639 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → (𝑠 <Q 𝑢𝑠 ∈ (1st𝐴)))
5511, 54sylan 271 . . . . . . . . . . . . . 14 ((𝐴P𝑢 ∈ (1st𝐴)) → (𝑠 <Q 𝑢𝑠 ∈ (1st𝐴)))
5652, 53, 55syl2anc 397 . . . . . . . . . . . . 13 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝑠 <Q 𝑢𝑠 ∈ (1st𝐴)))
5751, 56mpd 13 . . . . . . . . . . . 12 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠 ∈ (1st𝐴))
58 rspe 2387 . . . . . . . . . . . 12 ((𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴))) → ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴)))
5937, 38, 57, 58syl12anc 1144 . . . . . . . . . . 11 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴)))
6016adantr 265 . . . . . . . . . . . 12 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝐵P)
61 ltdfpr 6661 . . . . . . . . . . . 12 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴))))
6260, 52, 61syl2anc 397 . . . . . . . . . . 11 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝐵<P 𝐴 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴))))
6359, 62mpbird 160 . . . . . . . . . 10 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝐵<P 𝐴)
64 simpll3 956 . . . . . . . . . . 11 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → ¬ 𝐵<P 𝐴)
6564ad3antrrr 469 . . . . . . . . . 10 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → ¬ 𝐵<P 𝐴)
6663, 65pm2.21dd 560 . . . . . . . . 9 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑥 ∈ (2nd𝐴))
6766ex 112 . . . . . . . 8 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑥 <Q (𝑢 +Q 𝑡) → 𝑥 ∈ (2nd𝐴)))
68 simprr 492 . . . . . . . . 9 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑢 +Q 𝑡) ∈ (2nd𝐴))
69 eleq1 2116 . . . . . . . . 9 (𝑥 = (𝑢 +Q 𝑡) → (𝑥 ∈ (2nd𝐴) ↔ (𝑢 +Q 𝑡) ∈ (2nd𝐴)))
7068, 69syl5ibrcom 150 . . . . . . . 8 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑥 = (𝑢 +Q 𝑡) → 𝑥 ∈ (2nd𝐴)))
71 prcunqu 6640 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) <Q 𝑥𝑥 ∈ (2nd𝐴)))
7211, 71sylan 271 . . . . . . . . 9 ((𝐴P ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) <Q 𝑥𝑥 ∈ (2nd𝐴)))
7322, 68, 72syl2anc 397 . . . . . . . 8 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑢 +Q 𝑡) <Q 𝑥𝑥 ∈ (2nd𝐴)))
7467, 70, 733jaod 1210 . . . . . . 7 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥) → 𝑥 ∈ (2nd𝐴)))
7531, 74mpd 13 . . . . . 6 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑥 ∈ (2nd𝐴))
7614, 75rexlimddv 2454 . . . . 5 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝑥 ∈ (2nd𝐴))
777, 76rexlimddv 2454 . . . 4 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑥 ∈ (2nd𝐴))
784, 77rexlimddv 2454 . . 3 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝑥 ∈ (2nd𝐴))
7978ex 112 . 2 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (2nd𝐵) → 𝑥 ∈ (2nd𝐴)))
8079ssrdv 2978 1 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (2nd𝐵) ⊆ (2nd𝐴))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ↔ wb 102   ∨ w3o 895   ∧ w3a 896   = wceq 1259   ∈ wcel 1409  ∃wrex 2324   ⊆ wss 2944  ⟨cop 3405   class class class wbr 3791  ‘cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  Qcnq 6435   +Q cplq 6437
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