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Theorem recexprlemopu 6599
 Description: The upper cut of B is open. Lemma for recexpr 6610. (Contributed by Jim Kingdon, 28-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩
Assertion
Ref Expression
recexprlemopu ((A P 𝑟 Q 𝑟 (2ndB)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)))
Distinct variable groups:   𝑟,𝑞,x,y,A   B,𝑞,𝑟,x,y

Proof of Theorem recexprlemopu
StepHypRef Expression
1 recexpr.1 . . . 4 B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩
21recexprlemelu 6595 . . 3 (𝑟 (2ndB) ↔ y(y <Q 𝑟 (*Qy) (1stA)))
3 ltbtwnnqq 6398 . . . . . 6 (y <Q 𝑟𝑞 Q (y <Q 𝑞 𝑞 <Q 𝑟))
43biimpi 113 . . . . 5 (y <Q 𝑟𝑞 Q (y <Q 𝑞 𝑞 <Q 𝑟))
5 simplr 482 . . . . . . . 8 (((y <Q 𝑞 𝑞 <Q 𝑟) (*Qy) (1stA)) → 𝑞 <Q 𝑟)
6 19.8a 1479 . . . . . . . . . 10 ((y <Q 𝑞 (*Qy) (1stA)) → y(y <Q 𝑞 (*Qy) (1stA)))
71recexprlemelu 6595 . . . . . . . . . 10 (𝑞 (2ndB) ↔ y(y <Q 𝑞 (*Qy) (1stA)))
86, 7sylibr 137 . . . . . . . . 9 ((y <Q 𝑞 (*Qy) (1stA)) → 𝑞 (2ndB))
98adantlr 446 . . . . . . . 8 (((y <Q 𝑞 𝑞 <Q 𝑟) (*Qy) (1stA)) → 𝑞 (2ndB))
105, 9jca 290 . . . . . . 7 (((y <Q 𝑞 𝑞 <Q 𝑟) (*Qy) (1stA)) → (𝑞 <Q 𝑟 𝑞 (2ndB)))
1110expcom 109 . . . . . 6 ((*Qy) (1stA) → ((y <Q 𝑞 𝑞 <Q 𝑟) → (𝑞 <Q 𝑟 𝑞 (2ndB))))
1211reximdv 2414 . . . . 5 ((*Qy) (1stA) → (𝑞 Q (y <Q 𝑞 𝑞 <Q 𝑟) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB))))
134, 12mpan9 265 . . . 4 ((y <Q 𝑟 (*Qy) (1stA)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)))
1413exlimiv 1486 . . 3 (y(y <Q 𝑟 (*Qy) (1stA)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)))
152, 14sylbi 114 . 2 (𝑟 (2ndB) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)))
16153ad2ant3 926 1 ((A P 𝑟 Q 𝑟 (2ndB)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∃wrex 2301  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  *Qcrq 6268
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