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Theorem prcdnq 9853
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 9786 . . . . . . 7 <Q ⊆ (Q × Q)
2 relxp 5160 . . . . . . 7 Rel (Q × Q)
3 relss 5240 . . . . . . 7 ( <Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel <Q ))
41, 2, 3mp2 9 . . . . . 6 Rel <Q
54brrelexi 5192 . . . . 5 (𝐶 <Q 𝐵𝐶 ∈ V)
6 eleq1 2718 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
76anbi2d 740 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐴P𝑥𝐴) ↔ (𝐴P𝐵𝐴)))
8 breq2 4689 . . . . . . . 8 (𝑥 = 𝐵 → (𝑦 <Q 𝑥𝑦 <Q 𝐵))
97, 8anbi12d 747 . . . . . . 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 4688 . . . . . . . 8 (𝑦 = 𝐶 → (𝑦 <Q 𝐵𝐶 <Q 𝐵))
1211anbi2d 740 . . . . . . 7 (𝑦 = 𝐶 → (((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) ↔ ((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵)))
13 eleq1 2718 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝐴𝐶𝐴))
1412, 13imbi12d 333 . . . . . 6 (𝑦 = 𝐶 → ((((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦𝐴) ↔ (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴)))
15 elnpi 9848 . . . . . . . . . . 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 2961 . . . . . . . . 9 ((𝐴P𝑥𝐴) → (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))
1817simpld 474 . . . . . . . 8 ((𝐴P𝑥𝐴) → ∀𝑦(𝑦 <Q 𝑥𝑦𝐴))
191819.21bi 2097 . . . . . . 7 ((𝐴P𝑥𝐴) → (𝑦 <Q 𝑥𝑦𝐴))
2019imp 444 . . . . . 6 (((𝐴P𝑥𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦𝐴)
2110, 14, 20vtocl2g 3301 . . . . 5 ((𝐵𝐴𝐶 ∈ V) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
225, 21sylan2 490 . . . 4 ((𝐵𝐴𝐶 <Q 𝐵) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
2322adantll 750 . . 3 (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
2423pm2.43i 52 . 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 1054  wal 1521   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607  wpss 3608  c0 3948   class class class wbr 4685   × cxp 5141  Rel wrel 5148  Qcnq 9712   <Q cltq 9718  Pcnp 9719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-ltnq 9778  df-np 9841
This theorem is referenced by:  prub  9854  addclprlem1  9876  mulclprlem  9879  distrlem4pr  9886  1idpr  9889  psslinpr  9891  prlem934  9893  ltaddpr  9894  ltexprlem2  9897  ltexprlem3  9898  ltexprlem6  9901  prlem936  9907  reclem2pr  9908  suplem1pr  9912
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