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Theorem onelini 6504
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 6501 . 2 (𝐵𝐴𝐵𝐴)
3 dfss 3982 . 2 (𝐵𝐴𝐵 = (𝐵𝐴))
42, 3sylib 218 1 (𝐵𝐴𝐵 = (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cin 3962  wss 3963  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-in 3970  df-ss 3980  df-uni 4913  df-tr 5266  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by: (None)
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