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Theorem aptiprleml 7145
Description: Lemma for aptipr 7147. (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 6981 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prnmaxl 6994 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
31, 2sylan 277 . . . . . 6 ((𝐴P𝑥 ∈ (1st𝐴)) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
43ad2ant2rl 495 . . . . 5 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
5 ltexnqi 6915 . . . . . . 7 (𝑥 <Q 𝑠 → ∃𝑡Q (𝑥 +Q 𝑡) = 𝑠)
65ad2antll 475 . . . . . 6 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → ∃𝑡Q (𝑥 +Q 𝑡) = 𝑠)
7 simplr 497 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝐵P)
87ad2antrr 472 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝐵P)
9 simprl 498 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑡Q)
10 prop 6981 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
11 prarloc2 7010 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
1210, 11sylan 277 . . . . . . . 8 ((𝐵P𝑡Q) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
138, 9, 12syl2anc 403 . . . . . . 7 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
148adantr 270 . . . . . . . . . 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 498 . . . . . . . . . 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 6987 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑢 ∈ (1st𝐵)) → 𝑢Q)
1710, 16sylan 277 . . . . . . . . . 10 ((𝐵P𝑢 ∈ (1st𝐵)) → 𝑢Q)
1814, 15, 17syl2anc 403 . . . . . . . . 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 496 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝐴P)
2019ad3antrrr 476 . . . . . . . . . 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 499 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐴))
2221ad3antrrr 476 . . . . . . . . . 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 6987 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → 𝑥Q)
241, 23sylan 277 . . . . . . . . . 10 ((𝐴P𝑥 ∈ (1st𝐴)) → 𝑥Q)
2520, 22, 24syl2anc 403 . . . . . . . . 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 6902 . . . . . . . . 9 ((𝑢Q𝑥Q) → (𝑢 <Q 𝑥𝑢 = 𝑥𝑥 <Q 𝑢))
2718, 25, 26syl2anc 403 . . . . . . . 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 270 . . . . . . . . . . . . . 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 502 . . . . . . . . . . . . . . 15 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝑡Q)
3029adantr 270 . . . . . . . . . . . . . 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 6881 . . . . . . . . . . . . . 14 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
3228, 30, 31syl2anc 403 . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3433adantl 271 . . . . . . . . . . . . . . . . 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 6887 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3635adantl 271 . . . . . . . . . . . . . . . . 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 5773 . . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . . . . . . 19 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑠 ∈ (1st𝐴))
4039ad2antrr 472 . . . . . . . . . . . . . . . . . 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 2161 . . . . . . . . . . . . . . . . 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 6990 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑥 +Q 𝑡) ∈ (1st𝐴)) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st𝐴)))
431, 42sylan 277 . . . . . . . . . . . . . . . . 17 ((𝐴P ∧ (𝑥 +Q 𝑡) ∈ (1st𝐴)) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st𝐴)))
4420, 41, 43syl2anc 403 . . . . . . . . . . . . . . . 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 148 . . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . . 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 309 . . . . . . . . . . . . . 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 122 . . . . . . . . . . . . 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 2147 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (2nd𝐵) ↔ (𝑢 +Q 𝑡) ∈ (2nd𝐵)))
50 eleq1 2147 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (1st𝐴) ↔ (𝑢 +Q 𝑡) ∈ (1st𝐴)))
5149, 50anbi12d 457 . . . . . . . . . . . . . 14 (𝑣 = (𝑢 +Q 𝑡) → ((𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴)) ↔ ((𝑢 +Q 𝑡) ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st𝐴))))
5251rspcev 2715 . . . . . . . . . . . . 13 (((𝑢 +Q 𝑡) ∈ Q ∧ ((𝑢 +Q 𝑡) ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st𝐴))) → ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴)))
5332, 48, 52syl2anc 403 . . . . . . . . . . . 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 7012 . . . . . . . . . . . . . 14 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴))))
5514, 20, 54syl2anc 403 . . . . . . . . . . . . 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 270 . . . . . . . . . . . 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 165 . . . . . . . . . . 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 502 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → ¬ 𝐵<P 𝐴)
5958ad3antrrr 476 . . . . . . . . . . 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 583 . . . . . . . . . 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 113 . . . . . . . . 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 2147 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢 ∈ (1st𝐵) ↔ 𝑥 ∈ (1st𝐵)))
6315, 62syl5ibcom 153 . . . . . . . . 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 6990 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑢 ∈ (1st𝐵)) → (𝑥 <Q 𝑢𝑥 ∈ (1st𝐵)))
6510, 64sylan 277 . . . . . . . . . 10 ((𝐵P𝑢 ∈ (1st𝐵)) → (𝑥 <Q 𝑢𝑥 ∈ (1st𝐵)))
6614, 15, 65syl2anc 403 . . . . . . . . 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 1238 . . . . . . . 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 2489 . . . . . 6 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑥 ∈ (1st𝐵))
706, 69rexlimddv 2489 . . . . 5 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑥 ∈ (1st𝐵))
714, 70rexlimddv 2489 . . . 4 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐵))
7271expr 367 . . 3 (((𝐴P𝐵P) ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st𝐴) → 𝑥 ∈ (1st𝐵)))
73723impa 1136 . 2 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st𝐴) → 𝑥 ∈ (1st𝐵)))
7473ssrdv 3020 1 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (1st𝐴) ⊆ (1st𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  w3o 921  w3a 922   = wceq 1287  wcel 1436  wrex 2356  wss 2988  cop 3434   class class class wbr 3822  cfv 4983  (class class class)co 5615  1st c1st 5868  2nd c2nd 5869  Qcnq 6786   +Q cplq 6788   <Q cltq 6791  Pcnp 6797  <P cltp 6801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-eprel 4092  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-1o 6137  df-2o 6138  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-pli 6811  df-mi 6812  df-lti 6813  df-plpq 6850  df-mpq 6851  df-enq 6853  df-nqqs 6854  df-plqqs 6855  df-mqqs 6856  df-1nqqs 6857  df-rq 6858  df-ltnqqs 6859  df-enq0 6930  df-nq0 6931  df-0nq0 6932  df-plq0 6933  df-mq0 6934  df-inp 6972  df-iltp 6976
This theorem is referenced by:  aptipr  7147
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