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Theorem onelini 5877
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 5874 . 2 (𝐵𝐴𝐵𝐴)
3 dfss 3622 . 2 (𝐵𝐴𝐵 = (𝐵𝐴))
42, 3sylib 208 1 (𝐵𝐴𝐵 = (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cin 3606  wss 3607  Oncon0 5761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-uni 4469  df-tr 4786  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765
This theorem is referenced by: (None)
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