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Theorem onelini 6468
Description: An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
onelini (𝐵𝐴𝐵 = (𝐵𝐴))

Proof of Theorem onelini
StepHypRef Expression
1 on.1 . . 3 𝐴 ∈ On
21onelssi 6465 . 2 (𝐵𝐴𝐵𝐴)
3 dfss 3959 . 2 (𝐵𝐴𝐵 = (𝐵𝐴))
42, 3sylib 217 1 (𝐵𝐴𝐵 = (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cin 3940  wss 3941  Oncon0 6350
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-v 3472  df-in 3948  df-ss 3958  df-uni 4899  df-tr 5256  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6353  df-on 6354
This theorem is referenced by: (None)
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