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Theorem onelini 6285
 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 6282 . 2 (𝐵𝐴𝐵𝐴)
3 dfss 3936 . 2 (𝐵𝐴𝐵 = (𝐵𝐴))
42, 3sylib 221 1 (𝐵𝐴𝐵 = (𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115   ∩ cin 3917   ⊆ wss 3918  Oncon0 6174 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-v 3481  df-in 3925  df-ss 3935  df-uni 4822  df-tr 5156  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-ord 6177  df-on 6178 This theorem is referenced by: (None)
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