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Theorem predon 6938
Description: For an ordinal, the predecessor under E and On is an identity relationship. (Contributed by Scott Fenton, 27-Mar-2011.)
Assertion
Ref Expression
predon (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)

Proof of Theorem predon
StepHypRef Expression
1 predep 5665 . 2 (𝐴 ∈ On → Pred( E , On, 𝐴) = (On ∩ 𝐴))
2 onss 6937 . . 3 (𝐴 ∈ On → 𝐴 ⊆ On)
3 sseqin2 3795 . . 3 (𝐴 ⊆ On ↔ (On ∩ 𝐴) = 𝐴)
42, 3sylib 208 . 2 (𝐴 ∈ On → (On ∩ 𝐴) = 𝐴)
51, 4eqtrd 2655 1 (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cin 3554  wss 3555   E cep 4983  Predcpred 5638  Oncon0 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686
This theorem is referenced by:  dfrecs3  7414  tfr2ALT  7442  tfr3ALT  7443
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