MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elpredg Structured version   Visualization version   GIF version

Theorem elpredg 6213
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 6212 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
2 ibar 528 . . . 4 (𝑌𝐴 → (𝑌𝑅𝑋 ↔ (𝑌𝐴𝑌𝑅𝑋)))
32bicomd 222 . . 3 (𝑌𝐴 → ((𝑌𝐴𝑌𝑅𝑋) ↔ 𝑌𝑅𝑋))
43adantl 481 . 2 ((𝑋𝐵𝑌𝐴) → ((𝑌𝐴𝑌𝑅𝑋) ↔ 𝑌𝑅𝑋))
51, 4bitrd 278 1 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2109   class class class wbr 5078  Predcpred 6198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-xp 5594  df-cnv 5596  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199
This theorem is referenced by:  predpoirr  6233  predfrirr  6234  wfrlem10OLD  8133  wsuclem  33798  wsuclb  33801
  Copyright terms: Public domain W3C validator