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Theorem predpoirr 6209
Description: Given a partial ordering, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
Assertion
Ref Expression
predpoirr (𝑅 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))

Proof of Theorem predpoirr
StepHypRef Expression
1 poirr 5495 . . . . 5 ((𝑅 Po 𝐴𝑋𝐴) → ¬ 𝑋𝑅𝑋)
2 elpredg 6189 . . . . . . 7 ((𝑋𝐴𝑋𝐴) → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
32anidms 570 . . . . . 6 (𝑋𝐴 → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
43notbid 321 . . . . 5 (𝑋𝐴 → (¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ ¬ 𝑋𝑅𝑋))
51, 4syl5ibr 249 . . . 4 (𝑋𝐴 → ((𝑅 Po 𝐴𝑋𝐴) → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
65expd 419 . . 3 (𝑋𝐴 → (𝑅 Po 𝐴 → (𝑋𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))))
76pm2.43b 55 . 2 (𝑅 Po 𝐴 → (𝑋𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
8 predel 6196 . . 3 (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑋𝐴)
98con3i 157 . 2 𝑋𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
107, 9pm2.61d1 183 1 (𝑅 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2111   class class class wbr 5068   Po wpo 5481  Predcpred 6175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-br 5069  df-opab 5131  df-po 5483  df-xp 5572  df-cnv 5574  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-pred 6176
This theorem is referenced by:  xpord2ind  33560  xpord3ind  33566
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