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Theorem predpoirr 6344
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 5606 . . . . 5 ((𝑅 Po 𝐴𝑋𝐴) → ¬ 𝑋𝑅𝑋)
2 elpredg 6324 . . . . . . 7 ((𝑋𝐴𝑋𝐴) → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
32anidms 565 . . . . . 6 (𝑋𝐴 → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
43notbid 317 . . . . 5 (𝑋𝐴 → (¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ ¬ 𝑋𝑅𝑋))
51, 4imbitrrid 245 . . . 4 (𝑋𝐴 → ((𝑅 Po 𝐴𝑋𝐴) → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
65expd 414 . . 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 6331 . . 3 (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑋𝐴)
98con3i 154 . 2 𝑋𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
107, 9pm2.61d1 180 1 (𝑅 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wcel 2098   class class class wbr 5152   Po wpo 5592  Predcpred 6309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-po 5594  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310
This theorem is referenced by:  xpord2indlem  8160  xpord3inddlem  8167
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