Theorem predon 7805

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

Step	Hyp	Ref	Expression
1		tron 6409	. 2 ⊢ Tr On
2		trpred 6354	. 2 ⊢ ((Tr On ∧ 𝐴 ∈ On) → Pred( E , On, 𝐴) = 𝐴)
3	1, 2	mpan 690	1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Tr wtr 5265   E cep 5588  Predcpred 6322  Oncon0 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390

This theorem is referenced by:  dfrecs3  8411  dfrecs3OLD  8412  tfr2ALT  8440  tfr3ALT  8441  on2recsov  8705  on2ind  8706  on3ind  8707