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Theorem aptiprleml 6794
Description: Lemma for aptipr 6796. (Contributed by Jim Kingdon, 28-Jan-2020.)
Assertion
Ref Expression
aptiprleml ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (1st𝐴) ⊆ (1st𝐵))

Proof of Theorem aptiprleml
Dummy variables 𝑓 𝑔 𝑠 𝑡 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6630 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prnmaxl 6643 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
31, 2sylan 271 . . . . . 6 ((𝐴P𝑥 ∈ (1st𝐴)) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
43ad2ant2rl 488 . . . . 5 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
5 ltexnqi 6564 . . . . . . 7 (𝑥 <Q 𝑠 → ∃𝑡Q (𝑥 +Q 𝑡) = 𝑠)
65ad2antll 468 . . . . . 6 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → ∃𝑡Q (𝑥 +Q 𝑡) = 𝑠)
7 simplr 490 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝐵P)
87ad2antrr 465 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝐵P)
9 simprl 491 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑡Q)
10 prop 6630 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
11 prarloc2 6659 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
1210, 11sylan 271 . . . . . . . 8 ((𝐵P𝑡Q) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
138, 9, 12syl2anc 397 . . . . . . 7 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
148adantr 265 . . . . . . . . . 10 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝐵P)
15 simprl 491 . . . . . . . . . 10 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝑢 ∈ (1st𝐵))
16 elprnql 6636 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑢 ∈ (1st𝐵)) → 𝑢Q)
1710, 16sylan 271 . . . . . . . . . 10 ((𝐵P𝑢 ∈ (1st𝐵)) → 𝑢Q)
1814, 15, 17syl2anc 397 . . . . . . . . 9 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝑢Q)
19 simpll 489 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝐴P)
2019ad3antrrr 469 . . . . . . . . . 10 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝐴P)
21 simprr 492 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐴))
2221ad3antrrr 469 . . . . . . . . . 10 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝑥 ∈ (1st𝐴))
23 elprnql 6636 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → 𝑥Q)
241, 23sylan 271 . . . . . . . . . 10 ((𝐴P𝑥 ∈ (1st𝐴)) → 𝑥Q)
2520, 22, 24syl2anc 397 . . . . . . . . 9 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝑥Q)
26 nqtri3or 6551 . . . . . . . . 9 ((𝑢Q𝑥Q) → (𝑢 <Q 𝑥𝑢 = 𝑥𝑥 <Q 𝑢))
2718, 25, 26syl2anc 397 . . . . . . . 8 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝑢 <Q 𝑥𝑢 = 𝑥𝑥 <Q 𝑢))
2818adantr 265 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ 𝑢 <Q 𝑥) → 𝑢Q)
29 simplrl 495 . . . . . . . . . . . . . . 15 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝑡Q)
3029adantr 265 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ 𝑢 <Q 𝑥) → 𝑡Q)
31 addclnq 6530 . . . . . . . . . . . . . 14 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
3228, 30, 31syl2anc 397 . . . . . . . . . . . . 13 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ 𝑢 <Q 𝑥) → (𝑢 +Q 𝑡) ∈ Q)
33 ltanqg 6555 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3433adantl 266 . . . . . . . . . . . . . . . . 17 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
35 addcomnqg 6536 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3635adantl 266 . . . . . . . . . . . . . . . . 17 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3734, 18, 25, 29, 36caovord2d 5697 . . . . . . . . . . . . . . . 16 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝑢 <Q 𝑥 ↔ (𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡)))
38 simplrr 496 . . . . . . . . . . . . . . . . . 18 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝑥 +Q 𝑡) = 𝑠)
39 simprl 491 . . . . . . . . . . . . . . . . . . 19 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑠 ∈ (1st𝐴))
4039ad2antrr 465 . . . . . . . . . . . . . . . . . 18 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝑠 ∈ (1st𝐴))
4138, 40eqeltrd 2130 . . . . . . . . . . . . . . . . 17 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝑥 +Q 𝑡) ∈ (1st𝐴))
42 prcdnql 6639 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑥 +Q 𝑡) ∈ (1st𝐴)) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st𝐴)))
431, 42sylan 271 . . . . . . . . . . . . . . . . 17 ((𝐴P ∧ (𝑥 +Q 𝑡) ∈ (1st𝐴)) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st𝐴)))
4420, 41, 43syl2anc 397 . . . . . . . . . . . . . . . 16 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st𝐴)))
4537, 44sylbid 143 . . . . . . . . . . . . . . 15 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝑢 <Q 𝑥 → (𝑢 +Q 𝑡) ∈ (1st𝐴)))
46 simprr 492 . . . . . . . . . . . . . . 15 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝑢 +Q 𝑡) ∈ (2nd𝐵))
4745, 46jctild 303 . . . . . . . . . . . . . 14 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝑢 <Q 𝑥 → ((𝑢 +Q 𝑡) ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st𝐴))))
4847imp 119 . . . . . . . . . . . . 13 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ 𝑢 <Q 𝑥) → ((𝑢 +Q 𝑡) ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st𝐴)))
49 eleq1 2116 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (2nd𝐵) ↔ (𝑢 +Q 𝑡) ∈ (2nd𝐵)))
50 eleq1 2116 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (1st𝐴) ↔ (𝑢 +Q 𝑡) ∈ (1st𝐴)))
5149, 50anbi12d 450 . . . . . . . . . . . . . 14 (𝑣 = (𝑢 +Q 𝑡) → ((𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴)) ↔ ((𝑢 +Q 𝑡) ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st𝐴))))
5251rspcev 2673 . . . . . . . . . . . . 13 (((𝑢 +Q 𝑡) ∈ Q ∧ ((𝑢 +Q 𝑡) ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st𝐴))) → ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴)))
5332, 48, 52syl2anc 397 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ 𝑢 <Q 𝑥) → ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴)))
54 ltdfpr 6661 . . . . . . . . . . . . . 14 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴))))
5514, 20, 54syl2anc 397 . . . . . . . . . . . . 13 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝐵<P 𝐴 ↔ ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴))))
5655adantr 265 . . . . . . . . . . . 12 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ 𝑢 <Q 𝑥) → (𝐵<P 𝐴 ↔ ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴))))
5753, 56mpbird 160 . . . . . . . . . . 11 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ 𝑢 <Q 𝑥) → 𝐵<P 𝐴)
58 simplrl 495 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → ¬ 𝐵<P 𝐴)
5958ad3antrrr 469 . . . . . . . . . . 11 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ 𝑢 <Q 𝑥) → ¬ 𝐵<P 𝐴)
6057, 59pm2.21dd 560 . . . . . . . . . 10 (((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) ∧ 𝑢 <Q 𝑥) → 𝑥 ∈ (1st𝐵))
6160ex 112 . . . . . . . . 9 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝑢 <Q 𝑥𝑥 ∈ (1st𝐵)))
62 eleq1 2116 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢 ∈ (1st𝐵) ↔ 𝑥 ∈ (1st𝐵)))
6315, 62syl5ibcom 148 . . . . . . . . 9 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝑢 = 𝑥𝑥 ∈ (1st𝐵)))
64 prcdnql 6639 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑢 ∈ (1st𝐵)) → (𝑥 <Q 𝑢𝑥 ∈ (1st𝐵)))
6510, 64sylan 271 . . . . . . . . . 10 ((𝐵P𝑢 ∈ (1st𝐵)) → (𝑥 <Q 𝑢𝑥 ∈ (1st𝐵)))
6614, 15, 65syl2anc 397 . . . . . . . . 9 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → (𝑥 <Q 𝑢𝑥 ∈ (1st𝐵)))
6761, 63, 663jaod 1210 . . . . . . . 8 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → ((𝑢 <Q 𝑥𝑢 = 𝑥𝑥 <Q 𝑢) → 𝑥 ∈ (1st𝐵)))
6827, 67mpd 13 . . . . . . 7 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝑥 ∈ (1st𝐵))
6913, 68rexlimddv 2454 . . . . . 6 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑥 ∈ (1st𝐵))
706, 69rexlimddv 2454 . . . . 5 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑥 ∈ (1st𝐵))
714, 70rexlimddv 2454 . . . 4 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐵))
7271expr 361 . . 3 (((𝐴P𝐵P) ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st𝐴) → 𝑥 ∈ (1st𝐵)))
73723impa 1110 . 2 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st𝐴) → 𝑥 ∈ (1st𝐵)))
7473ssrdv 2978 1 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (1st𝐴) ⊆ (1st𝐵))
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   <Q cltq 6440  Pcnp 6446  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iltp 6625
This theorem is referenced by:  aptipr  6796
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