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

Theorem elpred 6317
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.) (Proof shortened by BJ, 16-Oct-2024.)
Hypothesis
Ref Expression
elpred.1 𝑌 ∈ V
Assertion
Ref Expression
elpred (𝑋𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))

Proof of Theorem elpred
StepHypRef Expression
1 elpred.1 . 2 𝑌 ∈ V
2 elpredgg 6313 . 2 ((𝑋𝐷𝑌 ∈ V) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
31, 2mpan2 689 1 (𝑋𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  Vcvv 3474   class class class wbr 5148  Predcpred 6299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300
This theorem is referenced by:  predtrss  6323  setlikespec  6326  preddowncl  6333  xpord2pred  8130  xpord3pred  8137  fprlem2  8285  wfrlem10OLD  8317  ttrclselem2  9720  ttrclse  9721  preduz  13622  predfz  13625  wzel  34791  wsuclem  34792
  Copyright terms: Public domain W3C validator