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Theorem predon 7486
Description: The predecessor of an ordinal under E and On is itself. (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 6142 . 2 (𝐴 ∈ On → Pred( E , On, 𝐴) = (On ∩ 𝐴))
2 onss 7485 . . 3 (𝐴 ∈ On → 𝐴 ⊆ On)
3 sseqin2 4142 . . 3 (𝐴 ⊆ On ↔ (On ∩ 𝐴) = 𝐴)
42, 3sylib 221 . 2 (𝐴 ∈ On → (On ∩ 𝐴) = 𝐴)
51, 4eqtrd 2833 1 (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  cin 3880  wss 3881   E cep 5429  Predcpred 6115  Oncon0 6159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163
This theorem is referenced by:  dfrecs3  7992  tfr2ALT  8020  tfr3ALT  8021
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