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Theorem predpoirr 6169
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 5478 . . . . 5 ((𝑅 Po 𝐴𝑋𝐴) → ¬ 𝑋𝑅𝑋)
2 elpredg 6155 . . . . . . 7 ((𝑋𝐴𝑋𝐴) → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
32anidms 569 . . . . . 6 (𝑋𝐴 → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
43notbid 320 . . . . 5 (𝑋𝐴 → (¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ ¬ 𝑋𝑅𝑋))
51, 4syl5ibr 248 . . . 4 (𝑋𝐴 → ((𝑅 Po 𝐴𝑋𝐴) → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
65expd 418 . . 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 6158 . . 3 (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑋𝐴)
98con3i 157 . 2 𝑋𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
107, 9pm2.61d1 182 1 (𝑅 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wcel 2107   class class class wbr 5057   Po wpo 5465  Predcpred 6140
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-po 5467  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141
This theorem is referenced by: (None)
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