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Theorem prcdnq 9760
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 9693 . . . . . . 7 <Q ⊆ (Q × Q)
2 relxp 5193 . . . . . . 7 Rel (Q × Q)
3 relss 5172 . . . . . . 7 ( <Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel <Q ))
41, 2, 3mp2 9 . . . . . 6 Rel <Q
54brrelexi 5123 . . . . 5 (𝐶 <Q 𝐵𝐶 ∈ V)
6 eleq1 2692 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
76anbi2d 739 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐴P𝑥𝐴) ↔ (𝐴P𝐵𝐴)))
8 breq2 4622 . . . . . . . 8 (𝑥 = 𝐵 → (𝑦 <Q 𝑥𝑦 <Q 𝐵))
97, 8anbi12d 746 . . . . . . 7 (𝑥 = 𝐵 → (((𝐴P𝑥𝐴) ∧ 𝑦 <Q 𝑥) ↔ ((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵)))
109imbi1d 331 . . . . . 6 (𝑥 = 𝐵 → ((((𝐴P𝑥𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦𝐴) ↔ (((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦𝐴)))
11 breq1 4621 . . . . . . . 8 (𝑦 = 𝐶 → (𝑦 <Q 𝐵𝐶 <Q 𝐵))
1211anbi2d 739 . . . . . . 7 (𝑦 = 𝐶 → (((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) ↔ ((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵)))
13 eleq1 2692 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝐴𝐶𝐴))
1412, 13imbi12d 334 . . . . . 6 (𝑦 = 𝐶 → ((((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦𝐴) ↔ (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴)))
15 elnpi 9755 . . . . . . . . . . 11 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
1615simprbi 480 . . . . . . . . . 10 (𝐴P → ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))
1716r19.21bi 2932 . . . . . . . . 9 ((𝐴P𝑥𝐴) → (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))
1817simpld 475 . . . . . . . 8 ((𝐴P𝑥𝐴) → ∀𝑦(𝑦 <Q 𝑥𝑦𝐴))
191819.21bi 2062 . . . . . . 7 ((𝐴P𝑥𝐴) → (𝑦 <Q 𝑥𝑦𝐴))
2019imp 445 . . . . . 6 (((𝐴P𝑥𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦𝐴)
2110, 14, 20vtocl2g 3261 . . . . 5 ((𝐵𝐴𝐶 ∈ V) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
225, 21sylan2 491 . . . 4 ((𝐵𝐴𝐶 <Q 𝐵) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
2322adantll 749 . . 3 (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
2423pm2.43i 52 . 2 (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴)
2524ex 450 1 ((𝐴P𝐵𝐴) → (𝐶 <Q 𝐵𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1992  wral 2912  wrex 2913  Vcvv 3191  wss 3560  wpss 3561  c0 3896   class class class wbr 4618   × cxp 5077  Rel wrel 5084  Qcnq 9619   <Q cltq 9625  Pcnp 9626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-ltnq 9685  df-np 9748
This theorem is referenced by:  prub  9761  addclprlem1  9783  mulclprlem  9786  distrlem4pr  9793  1idpr  9796  psslinpr  9798  prlem934  9800  ltaddpr  9801  ltexprlem2  9804  ltexprlem3  9805  ltexprlem6  9808  prlem936  9814  reclem2pr  9815  suplem1pr  9819
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