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

Theorem predon 7725
Description: The predecessor of an ordinal under E and On is itself. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by BJ, 16-Oct-2024.)
Assertion
Ref Expression
predon (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)

Proof of Theorem predon
StepHypRef Expression
1 tron 6345 . 2 Tr On
2 trpred 6290 . 2 ((Tr On ∧ 𝐴 ∈ On) → Pred( E , On, 𝐴) = 𝐴)
31, 2mpan 689 1 (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Tr wtr 5227   E cep 5541  Predcpred 6257  Oncon0 6322
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326
This theorem is referenced by:  dfrecs3  8323  dfrecs3OLD  8324  tfr2ALT  8352  tfr3ALT  8353  on2recsov  8619  on2ind  8620  on3ind  8621
  Copyright terms: Public domain W3C validator