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Theorem prcdnq 9694
 Description: A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prcdnq ((𝐴P𝐵𝐴) → (𝐶 <Q 𝐵𝐶𝐴))

Proof of Theorem prcdnq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 9627 . . . . . . 7 <Q ⊆ (Q × Q)
2 relxp 5150 . . . . . . 7 Rel (Q × Q)
3 relss 5129 . . . . . . 7 ( <Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel <Q ))
41, 2, 3mp2 9 . . . . . 6 Rel <Q
54brrelexi 5082 . . . . 5 (𝐶 <Q 𝐵𝐶 ∈ V)
6 eleq1 2676 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
76anbi2d 736 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐴P𝑥𝐴) ↔ (𝐴P𝐵𝐴)))
8 breq2 4587 . . . . . . . 8 (𝑥 = 𝐵 → (𝑦 <Q 𝑥𝑦 <Q 𝐵))
97, 8anbi12d 743 . . . . . . 7 (𝑥 = 𝐵 → (((𝐴P𝑥𝐴) ∧ 𝑦 <Q 𝑥) ↔ ((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵)))
109imbi1d 330 . . . . . 6 (𝑥 = 𝐵 → ((((𝐴P𝑥𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦𝐴) ↔ (((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦𝐴)))
11 breq1 4586 . . . . . . . 8 (𝑦 = 𝐶 → (𝑦 <Q 𝐵𝐶 <Q 𝐵))
1211anbi2d 736 . . . . . . 7 (𝑦 = 𝐶 → (((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) ↔ ((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵)))
13 eleq1 2676 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝐴𝐶𝐴))
1412, 13imbi12d 333 . . . . . 6 (𝑦 = 𝐶 → ((((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦𝐴) ↔ (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴)))
15 elnpi 9689 . . . . . . . . . . 11 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
1615simprbi 479 . . . . . . . . . 10 (𝐴P → ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))
1716r19.21bi 2916 . . . . . . . . 9 ((𝐴P𝑥𝐴) → (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))
1817simpld 474 . . . . . . . 8 ((𝐴P𝑥𝐴) → ∀𝑦(𝑦 <Q 𝑥𝑦𝐴))
191819.21bi 2047 . . . . . . 7 ((𝐴P𝑥𝐴) → (𝑦 <Q 𝑥𝑦𝐴))
2019imp 444 . . . . . 6 (((𝐴P𝑥𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦𝐴)
2110, 14, 20vtocl2g 3243 . . . . 5 ((𝐵𝐴𝐶 ∈ V) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
225, 21sylan2 490 . . . 4 ((𝐵𝐴𝐶 <Q 𝐵) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
2322adantll 746 . . 3 (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
2423pm2.43i 50 . 2 (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴)
2524ex 449 1 ((𝐴P𝐵𝐴) → (𝐶 <Q 𝐵𝐶𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874   class class class wbr 4583   × cxp 5036  Rel wrel 5043  Qcnq 9553
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