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Theorem elpredg 6268
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 6267 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
2 ibar 530 . . . 4 (𝑌𝐴 → (𝑌𝑅𝑋 ↔ (𝑌𝐴𝑌𝑅𝑋)))
32bicomd 222 . . 3 (𝑌𝐴 → ((𝑌𝐴𝑌𝑅𝑋) ↔ 𝑌𝑅𝑋))
43adantl 483 . 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 205  wa 397  wcel 2107   class class class wbr 5106  Predcpred 6253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254
This theorem is referenced by:  predpoirr  6288  predfrirr  6289  wfrlem10OLD  8265  wsuclem  34456  wsuclb  34459
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