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Theorem aptiprleml 7629
Description: Lemma for aptipr 7631. (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 7465 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prnmaxl 7478 . . . . . . 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 7399 . . . . . . 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 7465 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
11 prarloc2 7494 . . . . . . . . 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 7471 . . . . . . . . . . 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 7471 . . . . . . . . . . 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 7386 . . . . . . . . 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 7365 . . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . . . . . 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 7371 . . . . . . . . . . . . . . . . . 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 6038 . . . . . . . . . . . . . . . 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 2254 . . . . . . . . . . . . . . . . 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 7474 . . . . . . . . . . . . . . . . . 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 2240 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢 +Q 𝑡) → (𝑣 ∈ (2nd𝐵) ↔ (𝑢 +Q 𝑡) ∈ (2nd𝐵)))
50 eleq1 2240 . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . 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 7496 . . . . . . . . . . . . . 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 620 . . . . . . . . . 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 2240 . . . . . . . . . 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 7474 . . . . . . . . . . 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 1304 . . . . . . . 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 2599 . . . . . 6 (((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) ∧ (𝑡Q ∧ (𝑥 +Q 𝑡) = 𝑠)) → 𝑥 ∈ (1st𝐵))
706, 69rexlimddv 2599 . . . . 5 ((((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) ∧ (𝑠 ∈ (1st𝐴) ∧ 𝑥 <Q 𝑠)) → 𝑥 ∈ (1st𝐵))
714, 70rexlimddv 2599 . . . 4 (((𝐴P𝐵P) ∧ (¬ 𝐵<P 𝐴𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐵))
7271expr 375 . . 3 (((𝐴P𝐵P) ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st𝐴) → 𝑥 ∈ (1st𝐵)))
73723impa 1194 . 2 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (1st𝐴) → 𝑥 ∈ (1st𝐵)))
7473ssrdv 3161 1 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (1st𝐴) ⊆ (1st𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  w3o 977  w3a 978   = wceq 1353  wcel 2148  wrex 2456  wss 3129  cop 3594   class class class wbr 4000  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Qcnq 7270   +Q cplq 7272   <Q cltq 7275  Pcnp 7281  <P cltp 7285
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iltp 7460
This theorem is referenced by:  aptipr  7631
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