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Theorem onelini 6052
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 6049 . 2 (𝐵𝐴𝐵𝐴)
3 dfss 3784 . 2 (𝐵𝐴𝐵 = (𝐵𝐴))
42, 3sylib 210 1 (𝐵𝐴𝐵 = (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cin 3768  wss 3769  Oncon0 5941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-in 3776  df-ss 3783  df-uni 4629  df-tr 4946  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944  df-on 5945
This theorem is referenced by: (None)
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