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Theorem predfrirr 6236
Description: Given a well-founded relation, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)
Assertion
Ref Expression
predfrirr (𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))

Proof of Theorem predfrirr
StepHypRef Expression
1 frirr 5567 . . . . 5 ((𝑅 Fr 𝐴𝑋𝐴) → ¬ 𝑋𝑅𝑋)
2 elpredg 6215 . . . . . . 7 ((𝑋𝐴𝑋𝐴) → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
32anidms 567 . . . . . 6 (𝑋𝐴 → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
43notbid 318 . . . . 5 (𝑋𝐴 → (¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ ¬ 𝑋𝑅𝑋))
51, 4syl5ibr 245 . . . 4 (𝑋𝐴 → ((𝑅 Fr 𝐴𝑋𝐴) → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
65expd 416 . . 3 (𝑋𝐴 → (𝑅 Fr 𝐴 → (𝑋𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))))
76pm2.43b 55 . 2 (𝑅 Fr 𝐴 → (𝑋𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
8 predel 6222 . . 3 (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑋𝐴)
98con3i 154 . 2 𝑋𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
107, 9pm2.61d1 180 1 (𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wcel 2110   class class class wbr 5079   Fr wfr 5542  Predcpred 6200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-fr 5545  df-xp 5596  df-cnv 5598  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201
This theorem is referenced by:  frrlem12  8104  wfrlem14OLD  8144
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