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Theorem inton 6241
Description: The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
Assertion
Ref Expression
inton On = ∅

Proof of Theorem inton
StepHypRef Expression
1 0elon 6237 . 2 ∅ ∈ On
2 int0el 4898 . 2 (∅ ∈ On → On = ∅)
31, 2ax-mp 5 1 On = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  c0 4288   cint 4867  Oncon0 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-pw 4537  df-uni 4831  df-int 4868  df-tr 5164  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188
This theorem is referenced by: (None)
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