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Theorem predpoirr 5926
Description: Given a partial ordering, 𝑋 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 5244 . . . . 5 ((𝑅 Po 𝐴𝑋𝐴) → ¬ 𝑋𝑅𝑋)
2 elpredg 5912 . . . . . . 7 ((𝑋𝐴𝑋𝐴) → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
32anidms 563 . . . . . 6 (𝑋𝐴 → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
43notbid 310 . . . . 5 (𝑋𝐴 → (¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ ¬ 𝑋𝑅𝑋))
51, 4syl5ibr 238 . . . 4 (𝑋𝐴 → ((𝑅 Po 𝐴𝑋𝐴) → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
65expd 405 . . 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 5915 . . 3 (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑋𝐴)
98con3i 152 . 2 𝑋𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
107, 9pm2.61d1 173 1 (𝑅 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wcel 2157   class class class wbr 4843   Po wpo 5231  Predcpred 5897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-po 5233  df-xp 5318  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898
This theorem is referenced by: (None)
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