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Theorem aptiprleml 7259
 Description: Lemma for aptipr 7261. (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 7095 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prnmaxl 7108 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
31, 2sylan 278 . . . . . 6 ((𝐴P𝑥 ∈ (1st𝐴)) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
43ad2ant2rl 496 . . . . 5 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
5 ltexnqi 7029 . . . . . . 7 (𝑥 <Q 𝑠 → ∃𝑡Q (𝑥 +Q 𝑡) = 𝑠)
65ad2antll 476 . . . . . 6 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → ∃𝑡Q (𝑥 +Q 𝑡) = 𝑠)
7 simplr 498 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝐵P)
87ad2antrr 473 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝐵P)
9 simprl 499 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑡Q)
10 prop 7095 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
11 prarloc2 7124 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
1210, 11sylan 278 . . . . . . . 8 ((𝐵P𝑡Q) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
138, 9, 12syl2anc 404 . . . . . . 7 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
148adantr 271 . . . . . . . . . 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 499 . . . . . . . . . 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 7101 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑢 ∈ (1st𝐵)) → 𝑢Q)
1710, 16sylan 278 . . . . . . . . . 10 ((𝐵P𝑢 ∈ (1st𝐵)) → 𝑢Q)
1814, 15, 17syl2anc 404 . . . . . . . . 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 497 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝐴P)
2019ad3antrrr 477 . . . . . . . . . 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 500 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐴))
2221ad3antrrr 477 . . . . . . . . . 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 7101 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → 𝑥Q)
241, 23sylan 278 . . . . . . . . . 10 ((𝐴P𝑥 ∈ (1st𝐴)) → 𝑥Q)
2520, 22, 24syl2anc 404 . . . . . . . . 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 7016 . . . . . . . . 9 ((𝑢Q𝑥Q) → (𝑢 <Q 𝑥𝑢 = 𝑥𝑥 <Q 𝑢))
2718, 25, 26syl2anc 404 . . . . . . . 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 271 . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . 15 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝑡Q)
3029adantr 271 . . . . . . . . . . . . . 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 6995 . . . . . . . . . . . . . 14 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
3228, 30, 31syl2anc 404 . . . . . . . . . . . . 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 7020 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3433adantl 272 . . . . . . . . . . . . . . . . 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 7001 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3635adantl 272 . . . . . . . . . . . . . . . . 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 5828 . . . . . . . . . . . . . . . 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 504 . . . . . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . . . . . . 19 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑠 ∈ (1st𝐴))
4039ad2antrr 473 . . . . . . . . . . . . . . . . . 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 2165 . . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑥 +Q 𝑡) ∈ (1st𝐴)) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st𝐴)))
431, 42sylan 278 . . . . . . . . . . . . . . . . 17 ((𝐴P ∧ (𝑥 +Q 𝑡) ∈ (1st𝐴)) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st𝐴)))
4420, 41, 43syl2anc 404 . . . . . . . . . . . . . . . 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 149 . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . . 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 310 . . . . . . . . . . . . . 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 123 . . . . . . . . . . . . 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 2151 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (2nd𝐵) ↔ (𝑢 +Q 𝑡) ∈ (2nd𝐵)))
50 eleq1 2151 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (1st𝐴) ↔ (𝑢 +Q 𝑡) ∈ (1st𝐴)))
5149, 50anbi12d 458 . . . . . . . . . . . . . 14 (𝑣 = (𝑢 +Q 𝑡) → ((𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴)) ↔ ((𝑢 +Q 𝑡) ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st𝐴))))
5251rspcev 2723 . . . . . . . . . . . . 13 (((𝑢 +Q 𝑡) ∈ Q ∧ ((𝑢 +Q 𝑡) ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st𝐴))) → ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴)))
5332, 48, 52syl2anc 404 . . . . . . . . . . . 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 7126 . . . . . . . . . . . . . 14 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴))))
5514, 20, 54syl2anc 404 . . . . . . . . . . . . 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 271 . . . . . . . . . . . 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 166 . . . . . . . . . . 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 503 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → ¬ 𝐵<P 𝐴)
5958ad3antrrr 477 . . . . . . . . . . 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 586 . . . . . . . . . 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 114 . . . . . . . . 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 2151 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢 ∈ (1st𝐵) ↔ 𝑥 ∈ (1st𝐵)))
6315, 62syl5ibcom 154 . . . . . . . . 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 7104 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑢 ∈ (1st𝐵)) → (𝑥 <Q 𝑢𝑥 ∈ (1st𝐵)))
6510, 64sylan 278 . . . . . . . . . 10 ((𝐵P𝑢 ∈ (1st𝐵)) → (𝑥 <Q 𝑢𝑥 ∈ (1st𝐵)))
6614, 15, 65syl2anc 404 . . . . . . . . 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 1241 . . . . . . . 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 2494 . . . . . 6 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑥 ∈ (1st𝐵))
706, 69rexlimddv 2494 . . . . 5 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑥 ∈ (1st𝐵))
714, 70rexlimddv 2494 . . . 4 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐵))
7271expr 368 . . 3 (((𝐴P𝐵P) ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st𝐴) → 𝑥 ∈ (1st𝐵)))
73723impa 1139 . 2 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st𝐴) → 𝑥 ∈ (1st𝐵)))
7473ssrdv 3032 1 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (1st𝐴) ⊆ (1st𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ w3o 924   ∧ w3a 925   = wceq 1290   ∈ wcel 1439  ∃wrex 2361   ⊆ wss 3000  ⟨cop 3453   class class class wbr 3851  ‘cfv 5028  (class class class)co 5666  1st c1st 5923  2nd c2nd 5924  Qcnq 6900   +Q cplq 6902
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