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Theorem aptiprleml 7673
Description: Lemma for aptipr 7675. (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 7509 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prnmaxl 7522 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
31, 2sylan 283 . . . . . 6 ((𝐴P𝑥 ∈ (1st𝐴)) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
43ad2ant2rl 511 . . . . 5 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → ∃𝑠 ∈ (1st𝐴)𝑥 <Q 𝑠)
5 ltexnqi 7443 . . . . . . 7 (𝑥 <Q 𝑠 → ∃𝑡Q (𝑥 +Q 𝑡) = 𝑠)
65ad2antll 491 . . . . . 6 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → ∃𝑡Q (𝑥 +Q 𝑡) = 𝑠)
7 simplr 528 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝐵P)
87ad2antrr 488 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝐵P)
9 simprl 529 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑡Q)
10 prop 7509 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
11 prarloc2 7538 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
1210, 11sylan 283 . . . . . . . 8 ((𝐵P𝑡Q) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
138, 9, 12syl2anc 411 . . . . . . 7 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → ∃𝑢 ∈ (1st𝐵)(𝑢 +Q 𝑡) ∈ (2nd𝐵))
148adantr 276 . . . . . . . . . 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 529 . . . . . . . . . 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 7515 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑢 ∈ (1st𝐵)) → 𝑢Q)
1710, 16sylan 283 . . . . . . . . . 10 ((𝐵P𝑢 ∈ (1st𝐵)) → 𝑢Q)
1814, 15, 17syl2anc 411 . . . . . . . . 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 527 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝐴P)
2019ad3antrrr 492 . . . . . . . . . 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 531 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐴))
2221ad3antrrr 492 . . . . . . . . . 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 7515 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → 𝑥Q)
241, 23sylan 283 . . . . . . . . . 10 ((𝐴P𝑥 ∈ (1st𝐴)) → 𝑥Q)
2520, 22, 24syl2anc 411 . . . . . . . . 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 7430 . . . . . . . . 9 ((𝑢Q𝑥Q) → (𝑢 <Q 𝑥𝑢 = 𝑥𝑥 <Q 𝑢))
2718, 25, 26syl2anc 411 . . . . . . . 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 276 . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . 15 ((((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) ∧ (𝑢 ∈ (1st𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐵))) → 𝑡Q)
3029adantr 276 . . . . . . . . . . . . . 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 7409 . . . . . . . . . . . . . 14 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
3228, 30, 31syl2anc 411 . . . . . . . . . . . . 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 7434 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3433adantl 277 . . . . . . . . . . . . . . . . 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 7415 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3635adantl 277 . . . . . . . . . . . . . . . . 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 6070 . . . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . . . . . . 19 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑠 ∈ (1st𝐴))
4039ad2antrr 488 . . . . . . . . . . . . . . . . . 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 2266 . . . . . . . . . . . . . . . . 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 7518 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑥 +Q 𝑡) ∈ (1st𝐴)) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st𝐴)))
431, 42sylan 283 . . . . . . . . . . . . . . . . 17 ((𝐴P ∧ (𝑥 +Q 𝑡) ∈ (1st𝐴)) → ((𝑢 +Q 𝑡) <Q (𝑥 +Q 𝑡) → (𝑢 +Q 𝑡) ∈ (1st𝐴)))
4420, 41, 43syl2anc 411 . . . . . . . . . . . . . . . 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 150 . . . . . . . . . . . . . . 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 531 . . . . . . . . . . . . . . 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 316 . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . 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 2252 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (2nd𝐵) ↔ (𝑢 +Q 𝑡) ∈ (2nd𝐵)))
50 eleq1 2252 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (1st𝐴) ↔ (𝑢 +Q 𝑡) ∈ (1st𝐴)))
5149, 50anbi12d 473 . . . . . . . . . . . . . 14 (𝑣 = (𝑢 +Q 𝑡) → ((𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴)) ↔ ((𝑢 +Q 𝑡) ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st𝐴))))
5251rspcev 2856 . . . . . . . . . . . . 13 (((𝑢 +Q 𝑡) ∈ Q ∧ ((𝑢 +Q 𝑡) ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (1st𝐴))) → ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴)))
5332, 48, 52syl2anc 411 . . . . . . . . . . . 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 7540 . . . . . . . . . . . . . 14 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ∃𝑣Q (𝑣 ∈ (2nd𝐵) ∧ 𝑣 ∈ (1st𝐴))))
5514, 20, 54syl2anc 411 . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 167 . . . . . . . . . . 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 535 . . . . . . . . . . . 12 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → ¬ 𝐵<P 𝐴)
5958ad3antrrr 492 . . . . . . . . . . 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 621 . . . . . . . . . 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 115 . . . . . . . . 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 2252 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢 ∈ (1st𝐵) ↔ 𝑥 ∈ (1st𝐵)))
6315, 62syl5ibcom 155 . . . . . . . . 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 7518 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑢 ∈ (1st𝐵)) → (𝑥 <Q 𝑢𝑥 ∈ (1st𝐵)))
6510, 64sylan 283 . . . . . . . . . 10 ((𝐵P𝑢 ∈ (1st𝐵)) → (𝑥 <Q 𝑢𝑥 ∈ (1st𝐵)))
6614, 15, 65syl2anc 411 . . . . . . . . 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 1315 . . . . . . . 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 2612 . . . . . 6 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑥 ∈ (1st𝐵))
706, 69rexlimddv 2612 . . . . 5 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑥 ∈ (1st𝐵))
714, 70rexlimddv 2612 . . . 4 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐵))
7271expr 375 . . 3 (((𝐴P𝐵P) ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st𝐴) → 𝑥 ∈ (1st𝐵)))
73723impa 1196 . 2 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st𝐴) → 𝑥 ∈ (1st𝐵)))
7473ssrdv 3176 1 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (1st𝐴) ⊆ (1st𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  w3o 979  w3a 980   = wceq 1364  wcel 2160  wrex 2469  wss 3144  cop 3613   class class class wbr 4021  cfv 5238  (class class class)co 5900  1st c1st 6167  2nd c2nd 6168  Qcnq 7314   +Q cplq 7316   <Q cltq 7319  Pcnp 7325  <P cltp 7329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-eprel 4310  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-irdg 6399  df-1o 6445  df-2o 6446  df-oadd 6449  df-omul 6450  df-er 6563  df-ec 6565  df-qs 6569  df-ni 7338  df-pli 7339  df-mi 7340  df-lti 7341  df-plpq 7378  df-mpq 7379  df-enq 7381  df-nqqs 7382  df-plqqs 7383  df-mqqs 7384  df-1nqqs 7385  df-rq 7386  df-ltnqqs 7387  df-enq0 7458  df-nq0 7459  df-0nq0 7460  df-plq0 7461  df-mq0 7462  df-inp 7500  df-iltp 7504
This theorem is referenced by:  aptipr  7675
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