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Theorem recexprlemlol 6597
 Description: The lower cut of B is lower. Lemma for recexpr 6609. (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
recexprlemlol ((A P 𝑞 Q) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)) → 𝑞 (1stB)))
Distinct variable groups:   𝑟,𝑞,x,y,A   B,𝑞,𝑟,x,y

Proof of Theorem recexprlemlol
StepHypRef Expression
1 ltsonq 6382 . . . . . . . . 9 <Q Or Q
2 ltrelnq 6349 . . . . . . . . 9 <Q ⊆ (Q × Q)
31, 2sotri 4663 . . . . . . . 8 ((𝑞 <Q 𝑟 𝑟 <Q y) → 𝑞 <Q y)
43ex 108 . . . . . . 7 (𝑞 <Q 𝑟 → (𝑟 <Q y𝑞 <Q y))
54anim1d 319 . . . . . 6 (𝑞 <Q 𝑟 → ((𝑟 <Q y (*Qy) (2ndA)) → (𝑞 <Q y (*Qy) (2ndA))))
65eximdv 1757 . . . . 5 (𝑞 <Q 𝑟 → (y(𝑟 <Q y (*Qy) (2ndA)) → y(𝑞 <Q y (*Qy) (2ndA))))
7 recexpr.1 . . . . . 6 B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩
87recexprlemell 6593 . . . . 5 (𝑟 (1stB) ↔ y(𝑟 <Q y (*Qy) (2ndA)))
97recexprlemell 6593 . . . . 5 (𝑞 (1stB) ↔ y(𝑞 <Q y (*Qy) (2ndA)))
106, 8, 93imtr4g 194 . . . 4 (𝑞 <Q 𝑟 → (𝑟 (1stB) → 𝑞 (1stB)))
1110imp 115 . . 3 ((𝑞 <Q 𝑟 𝑟 (1stB)) → 𝑞 (1stB))
1211rexlimivw 2423 . 2 (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)) → 𝑞 (1stB))
1312a1i 9 1 ((A P 𝑞 Q) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)) → 𝑞 (1stB)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = 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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