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Theorem elpredg 6335
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) (Proof shortened by BJ, 16-Oct-2024.)
Assertion
Ref Expression
elpredg ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))

Proof of Theorem elpredg
StepHypRef Expression
1 elpredgg 6334 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
2 ibar 528 . . . 4 (𝑌𝐴 → (𝑌𝑅𝑋 ↔ (𝑌𝐴𝑌𝑅𝑋)))
32bicomd 223 . . 3 (𝑌𝐴 → ((𝑌𝐴𝑌𝑅𝑋) ↔ 𝑌𝑅𝑋))
43adantl 481 . 2 ((𝑋𝐵𝑌𝐴) → ((𝑌𝐴𝑌𝑅𝑋) ↔ 𝑌𝑅𝑋))
51, 4bitrd 279 1 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108   class class class wbr 5143  Predcpred 6320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321
This theorem is referenced by:  predpoirr  6354  predfrirr  6355  wfrlem10OLD  8358  wsuclem  35826  wsuclb  35829
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