MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  predonOLD Structured version   Visualization version   GIF version

Theorem predonOLD 7778
Description: Obsolete version of predon 7777 as of 16-Oct-2024. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
predonOLD (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)

Proof of Theorem predonOLD
StepHypRef Expression
1 predep 6331 . 2 (𝐴 ∈ On → Pred( E , On, 𝐴) = (On ∩ 𝐴))
2 onss 7776 . . 3 (𝐴 ∈ On → 𝐴 ⊆ On)
3 sseqin2 4215 . . 3 (𝐴 ⊆ On ↔ (On ∩ 𝐴) = 𝐴)
42, 3sylib 217 . 2 (𝐴 ∈ On → (On ∩ 𝐴) = 𝐴)
51, 4eqtrd 2771 1 (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  cin 3947  wss 3948   E cep 5579  Predcpred 6299  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator