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Theorem predon 6883
 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 5623 . 2 (𝐴 ∈ On → Pred( E , On, 𝐴) = (On ∩ 𝐴))
2 onss 6882 . . 3 (𝐴 ∈ On → 𝐴 ⊆ On)
3 sseqin2 3779 . . 3 (𝐴 ⊆ On ↔ (On ∩ 𝐴) = 𝐴)
42, 3sylib 207 . 2 (𝐴 ∈ On → (On ∩ 𝐴) = 𝐴)
51, 4eqtrd 2644 1 (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540   E cep 4947  Predcpred 5596  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644 This theorem is referenced by:  dfrecs3  7356  tfr2ALT  7384  tfr3ALT  7385
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