Theorem predon 7789

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 6378	. 2 ⊢ Tr On
2		trpred 6327	. 2 ⊢ ((Tr On ∧ 𝐴 ∈ On) → Pred( E , On, 𝐴) = 𝐴)
3	1, 2	mpan 703	1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Tr wtr 5212   E cep 5550  Predcpred 6296  Oncon0 6355

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359

This theorem is used by:  dfrecs3  8364  tfr2ALT  8393  tfr3ALT  8394  on2recsov  8661  on2ind  8662  on3ind  8663